Write as many TRIG IDENTITIES As you can remember

Write as many TRIG IDENTITIES As you can remember
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Explore trigonometric identities, derive them from first principles, rearrange expressions, and prove various trigonometric equations. Enhance your understanding of trigonometry by solving equations and manipulating trig expressions fluently.

  • Trigonometry
  • Identities
  • Proofs
  • Equations
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  1. Starter Write as many TRIG IDENTITIES As you can remember How many can you derive from first principles ?

  2. Starter Rearrange into single expressions: 4 3 1+ x 3 x 1 1 + sin cos cot + cos ec

  3. Think Pair Share Trig: More proofs KUS objectives Be able to derive the trig identities Apply them when appropriate to solve trig equations Find exact values (addition formulas) Apply where appropriate to complete a Proof or show that question manipulate trig expressions fluently

  4. Qs I Nice Cannot be serious! Tricksy 9 Find without a calculator the exact value of 15 tan2 1 Solve for- < cos 8 ec 5 Solve for0 < 360 tan 8 + = 2 28 sec 1 = 3 3 Answer Answer Answer Answer Answer Answer 2 Prove 10 Prove 6 Prove sin + 1 cos A A sin cos ec ec = + = = sec2 sin tan cos sec 1 cos sin A A cos Answer Answer Hint Answer Answer Answer Hint Answer 3 Prove cos 7 Prove 11 Prove x (sec = cos sin ec cos sin + + 2 2 2 2 tan + )(cos cot ) x ec x x + = + sin cos 1 tan 1 cot cot = 2 2 1 2 sec cos x ec x Answer Hint Answer Hint Answer Answer Hint Answer Hint Answer 4 Prove (sin + 8 Prove 12 Prove sin + + 2 cos ) + 1 cos 2 A A + + + 2 1 ( = sin cos ) + = = 2 (sin cos ) 2 1 cos sin sin A A A + 1 ( 2 sin )( 1 cos ) Answer Answer Answer Answer Answer Answer

  5. Qs II Nice Cannot be serious! 21 Prove Tricksy 17 Prove 13 Prove = 2 2 tan cot sin 2 x 2 = + cos sin A A cos 2 A = cot x )(sec ec + cos sin A A (sec cos cos ) ec 1 cos x Answer Answer Hint Answer Answer Answer Hint Answer 14 Prove ( 18 Prove 22 Prove ) 1 1 + 2 2 + 2 tan x sin cos ec cos sin cos sin A A A A = 2 2 = = + + 2 2 2 sec 3 tan x x sin cot 3 2 sin Answer sec 2 A A Answer Answer Answer Answer Answer 19 Prove 23 Prove 15 Prove cot ( ) cot tan + + + tan tan sec + )(cot x cos x x x ecx = 2 2 cot = + = 1 ( sec )( 1 cos ) ecx sec cos ec Answer Answer Answer Answer Answer Answer 24 Prove 16 Prove 20 Prove ( ) ( ) + 2 2 4 4 2 2 tan cot sec tan x x + + 1 sin 1 sin x x cos = cos x 2+ x = + = 2 2 2 sec cos 2 ec 2 sec 1 x 2 4 tan x Answer Answer Answer Answer Answer Answer

  6. Exam Q1 8cosec2x=2/3 Solve for- < = 8 cos 3 ec 28 3 = cos 3 ec 28 24 = = sin 3 6 24 28 7 = 239 3 ... 301 , , 59 , 121 , 419 , 481 , ... = ... 100 , 3 . 79 , 7 . 19 , 7 . 40 , 3 . 139 , 7 . 160 , 3 . ... QED Back to Qs I Back to Qs I

  7. Exam Q2 sinxtanx+cosx=secx + = sin tan cos sec Prove: = tan sin cos cot A 2 + + = ec cos sin A cos sec cos A 2 2 sin tan 1 + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 cos cos sin sin cos cos = + LHS 2 2 = A 2 2 cos + 2 2 sin cos = 2 2 LHS cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 2 1 A 1 = LHS 2 1 1 2 2 cos + 2 cos cos 2 A A 1 1 2 2 = sec LHS QED Back to Qs I Back to Qs I

  8. Exam Q3 cosecx-sinx=cotxcosx = cos sin cot cos ec Prove: = tan sin cos cot A 2 + + = ec cos sin A cos sec cos A 2 2 sin tan 1 + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 sin 1 sin sin = LHS 2 2 sin = A 2 2 2 1 sin = LHS 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A sin 2 2 2 1 A cos2 2 = 1 1 LHS 2 2 sin + 2 cos cos 2 A A 1 1 2 2 cos = = cos cot cos LHS QED sin Back to Qs I Back to Qs I

  9. Exam Q4 (sinx+cosx)2 +(sinx-cosx)2=2 + + = 2 2 (sin cos ) (sin cos ) 2 Prove: = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 = sin + sin cos + 2 2 sin + 2 sin cos cos + LHS 2 2 2 2 2 cos = A 2 2 1 cot A 2 cos A ec cos sin A cos = + 2 2 2 2 (sin 2 cos ) LHS cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 2 1 A = 2 LHS QED 2 1 1 2 2 + 2 cos cos 2 A A 1 1 2 2 Back to Qs I Back to Qs I

  10. Exam Q5 8tan2x+secx=1 + = 2 8 tan sec 1 Solve for 0 < 360 the equation Giving your answers to 1 dp = tan sin cos cot A 2 + + = ec cos sin A cos sec cos A 2 2 sin tan 1 + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 ) 1 + = 2 (sec 8 sec 1 2 2 = A 2 2 + = 2 8 sec sec 9 0 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A + ) 1 = 8 ( sec )(sec 9 0 2 2 2 1 A = = sec sec 1 9 2 1 1 8 2 2 + 2 cos cos 2 A A 1 1 = = cos cos 1 2 2 8 9 = = 152 , 7 . 207 3 . 0 Back to Qs I Back to Qs I

  11. Exam Q6 sec2x=cosecx/(cosecx-sinx) sin cos ec ec = sec2 Prove: cos = cos (cos sin ) RHS ec ec 1 1 = 1 sin RHS sin sin 2 1 sin = RHS 2 sin sin 1 sin = RHS sin 1 sin 1 1 = = = 2 sec RHS QED 2 2 1 sin cos Back to Qs I Back to Qs I

  12. Exam Q7 cosx/(1-tanx) + sinx/(1-cotx) = sinx + cosx HINTS cos sin To Prove: + = + sin cos 1 tan 1 cot 1 Make the denominators fractions into and add/subtract together cos sin sin cos = = = = 1 tan ... 1 cot ... sin sin cos cos 2 by a fraction is equivalent to multiplying by the reciprocal = ( ) ( ) a b b a 3 If you want to make denominators the same = + 2 2 4 Difference of squares ( )( ) a b a b a b Next page for answers Back to Qs I Back to Qs I

  13. Exam Q7 cosx/(1-tanx) + sinx/(1-cotx) = sinx + cosx cos sin Prove: + = + sin cos 1 tan 1 cot Make into fractions sin cos cos sin = + LHS cos sin cos cos sin sin sin cos 2 2 cos sin = + LHS ( ) ( ) cos sin ( )( ) Difference of squares + 2 2 cos sin cos sin cos sin = = LHS ( ) ( ) cos sin cos sin = cos + sin QED Back to Qs I Back to Qs I

  14. Exam Q8 (1+sinx+cosx)2 = 2(1+sinx)(1+cosx) + + = + + 2 1 ( sin cos ) 1 ( 2 sin )( 1 cos ) Prove: = + + + + + 2 2 1 2 sin 2 cos 2 sin cos sin cos LHS = + + + + 1 2 sin 2 cos 2 sin cos 1 LHS = tan sin cos cot A 2 + + = ec cos sin A cos sec cos A 2 2 sin tan 1 + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 1 ( 2 = + + + sin cos sin cos ) LHS 2 2 = A 2 2 1 ( 2 = + + sin )( 1 cos ) LHS QED 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 2 1 A 2 1 1 2 2 + 2 cos cos 2 A A 1 1 2 2 Back to Qs I Back to Qs I

  15. Exam Q9 find tan2 15 tan2 15 Find without a calculator the exact simplified value of tan + tan A B = tan( ) A B tan + 45 tan 30 1 tan tan A B = tan( 45 30 ) 1 tan 45 tan 30 3 3 1 3 3 3 = = = 3 3 tan( 15 ) + 1 1 ( )( ) 3 + + 3 3 3 3 3 3 + 9 3 3 3 3 3 12 6 3 2 3 ( ) 2 = = = tan 15 + + + + + 9 3 3 3 3 3 12 6 3 2 3 Back to Qs I Back to Qs I

  16. Exam Q10 sinx/(1+cosx)=(1-cosx)/sinx HINTS sin + 1 cos A A To Prove: = 1 cos sin A A 1 Multiply the LHS by (1 cosA) in numerator and denominator ( )( ) ... = A + 1 cos 1 cos A + = 2 2 2 Use sin cos 1 3 What cancels out Next page for answers Back to Qs I Back to Qs I

  17. Exam Q10 sinx/(1+cosx)=(1-cosx)/sinx = tan sin cos cot A 2 + + = ec cos sin A cos sec cos A 2 2 sin tan 1 + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 sin + 1 cos A A Prove: = 2 2 1 cos sin A A = A 2 2 sin + 1 cos A A 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A = LHS 1 cos 1 cos A A 2 2 2 1 A sin sin cos A 1 A 2 A 2 1 1 = LHS 2 2 cos A + 2 cos cos 2 A A 1 1 2 2 sin sin cos A A A = LHS 2 sin A 1 cos A = LHS sin A QED Back to Qs I Back to Qs I

  18. Exam Q11 (sec2x+tan2x)(cosec2x+cot2x)=1+2sec2xcosec2xHINTS To Prove: + + = + 2 2 2 2 2 2 (sec tan )(cos cot ) 1 2 sec cos x x ec x x x ec x 1 Multiply out = 2 2 2 Deal with tan cot ... x x 2 2 tan x 3 NOW use the trig identities for and cot x Next page for answers Back to Qs I Back to Qs I

  19. Exam Q11 (sec2x+tan2x)(cosec2x+cot2x)=1+2sec2xcosec2x + + = + 2 2 2 2 2 2 (sec tan )(cos cot ) 1 2 sec cos x x ec x x x ec x Prove: = + + + 2 2 2 2 2 2 2 2 sec cos (1) tan cos sec cot tan cot (3) LHS x ec x x ec x (2) x x x x ( ) ( ) 1 = + 2 2 2 2 ) 2 ( sec 1 cos sec cos x ec x x ec x = + 2 2 2 2 2 2 ) 2 ( sec cos cos sec cos sec x ec x ec (4) x x ec x x (4) + cos 2 2 1 1 cos sin 1 x x = + = = ) 4 ( 2 2 2 2 2 2 sin sec cos ec sin sin cos x x x x x x x x 2 2 sin cos x x = 2 2 ) 4 ( cos = = ) 3 ( 1 2 2 cos sin x x = + + 2 2 2 2 2 2 sec cos 2 sec cos sec cos 1 (3) LHS x ec x x ec (2) x x ec x QED (1) Back to Qs I Back to Qs I

  20. Exam Q12 + sin + 1 cos 2 A A + = Prove: 1 cos sin sin A A A + sin + 1 cos 1 cos sin A A A A = + LHS 1 cos 1 cos sin sin A A A A + sin sin cos sin cos sin A 1 A 2 A A A A = + LHS 2 cos sin A A + + sin sin cos sin cos sin A A A sin A A A A = LHS 2 A 2 sin 2 = = LHS 2 sin sin A A QED Back to Qs I Back to Qs I

  21. Second Set qs

  22. Exam Q13 tan2x cot2x = Prove = )(sec ec + 2 2 tan cot (sec cos cos ) ec = tan sin cos = ) 1 ) 1 2 2 (sec (cos LHS ec + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 2 2 = + 2 2 sec 1 cos 1 LHS ec = A 2 2 1 cot A 2 cos A ec cos sin A cos LHS = 2 2 sec cos ec 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 = )(sec ec + (sec cos cos ) LHS ec 2 1 A 2 1 1 2 2 QED + 2 cos cos 2 A A 1 1 2 2 Back to Qs II Back to Qs II

  23. Exam Q14 (sinx+cosecx)2=sin2x+cot2x+3 ( cos sin + ) 2 = + + 2 2 sin cot 3 ec Prove: = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 2 2 = + + 2 2 sin 2 sin cos cos LHS ec ec = A 2 2 1 cot A 2 cos A ec cos sin A cos 1 = + + + 2 2 sin 2 sin 1 ( cot ) LHS sin 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 2 1 A = + + 2 2 sin cot 3 LHS 2 1 1 QED 2 2 + 2 cos cos 2 A A 1 1 2 2 Back to Qs II Back to Qs II

  24. Exam Q15 cotx+tanx=secxcosecx + = cot tan sec cos ec Prove: = tan sin cos + + sec = 2 2 cos sin sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 = + LHS 2 2 sin cos = A 2 2 1 cot A 2 cos A ec cos sin A cos + 2 2 cos sin 2 2 = cos cos cos sin 2 2 2 cos sin A 2 A A A A A A LHS sin cos 2 2 2 1 A 1 = LHS 2 1 1 2 2 cos sin + 2 cos cos 2 A A 1 1 2 2 1 1 = = sec cos LHS ec sin cos QED Back to Qs II Back to Qs II

  25. Exam Q16 tan2x+cot2x=sec2x+cosec2x -2 + = + 2 2 2 2 tan cot sec cos 2 ec Prove: = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 = ) 1 + ) 1 2 2 (sec (cos LHS ec 2 2 = A 2 2 1 cot A 2 cos A ec cos sin A cos = + 2 2 sec cos 2 LHS ec QED 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 2 1 A 2 1 1 2 2 + 2 cos cos 2 A A 1 1 2 2 Back to Qs II Back to Qs II

  26. Exam Q17 sin 2 x 2 Prove: = cot x = 1 cos x tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 2 2 2 sin cos x x = A 2 2 1 cot A 2 cos A ec cos sin A cos LHS = 2x 1 1 ( 2 sin ) 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 sin cos x x 2 2 = 2 2 sin x 2 1 A 2 1 1 2 2 cos x + 2 cos cos 2 A A 1 1 = = cot x 2 2 sin x QED Back to Qs II Back to Qs II

  27. Exam Q18 2-tan2x = = 2 2 2 2 tan 2 sec 3 tan x x x Prove: = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 = + 2 2 1 ( 2 tan ) 3 tan RHS x x 2 2 = A 2 2 1 cot A 2 cos A ec cos sin A cos = + 2 2 2 2 tan 3 tan RHS x x 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 = 2 2 tan RHS x 2 1 A 2 1 1 QED 2 2 + 2 cos cos 2 A A 1 1 2 2 Back to Qs II Back to Qs II

  28. Exam Q19 Prove: ( tan ) + + = + + sec )(cot x cos 1 ( sec )( 1 cos ) x x ecx x ecx LHS= tan + + + cot sec cot tan cos sec cos x x x x x ecx x ecx = + + + sin cos cos sin x x x x 1 1 1 1 cos sin cos sin cos sin cos sin x x x x x x x x = + + + 1 cos sec sec cos ecx x x ecx = + + 1 ( sec )( 1 cos ) x ecx QED Back to Qs II Back to Qs II

  29. Exam Q20 = 4 4 2 sec tan 2 sec 1 x x x Prove: = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 2 2 2 2 sec sec tan tan x x x x LHS= 2 2 = A 2 2 1 cot A 2 cos A ec cos sin A cos = + ) 1 2 2 2 2 sec 1 ( x tan ) tan (sec x x x 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A = + + 2 2 2 2 2 2 sec sec tan tan sec tan x x x x x x 2 2 2 1 A = = = + + 2 2 sec sec tan (sec x x x 2 1 1 2 2 + 2 ) 1 cos cos 2 A A 2 2 1 1 x 2 2 2 sec 1 1 x QED Back to Qs II Back to Qs II

  30. Exam Q21 = + cos sin A A cos 2 A Prove: cos sin A A = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 + cos 2 cos sin A A A 2 2 LHS = + cos sin cos sin A A A A = A 2 2 1 cot A 2 cos A ec cos sin A cos + cos 2 (cos sin ) = A A sin ) sin A 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 cos A A 2 2 ( 2 1 A 2 2 sin + = cos (cos sin ) A A A A 2 1 1 2 2 2 2 cos A sin + A A + 2 cos cos 2 A A 1 1 2 2 = cos A QED Back to Qs II Back to Qs II

  31. = Exam Q22 tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 Prove 2 2 = 2 sin sec 2 A A 1 1 + = A 2 2 1 cot A 2 cos A ec cos sin A cos cos sin cos sin A A A A 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A LHS = 2 2 + cos sin cos sin A A A A 1 1 + 2 1 A + cos sin cos sin cos sin cos sin A A A A A A A A 2 1 1 2 2 + 2 cos cos 2 A A 1 1 2 2 = + sin cos sin cos A A 2 A A 2 2 2 sin sin cos cos A A A A = = 2 sin A sin = 2 2 2 cos sin 2 A A A = sin 2 sin sec 2 A A A 1 cos 2 cos 2 A A QED Back to Qs II Back to Qs II

  32. Exam Q23 = 2 cot tan 2 cot Prove: 2 2 tan 2 = 2 RHS= 2 tan 1 tan 1 2 tan = 2 2 tan 2 1 tan = 2 2 tan 2 tan ( ) = = 2 cot tan cot tan 1 1 2 2 QED Back to Qs II Back to Qs II

  33. Exam Q24 ( ) ( ) Prove: 2 2 + = 2+ 2 + sin 4 tan x 1 sin 1 x x cos cos x x = tan sin cos + + sec = 2 2 sin tan + sin cos 1 = sin 2 = = 1 = cos 2 = = = 1 2 2 ( ) ( ) 2 2 + + sin sin x x 1 1 LHS= = A 2 2 cos cos cos ( sec cos x x x x 1 cot A 2 cos A ec cos sin A cos ( ) )2 2 = + + sec tan tan x x x x 2 2 cos cos cos sin 2 2 2 cos sin A 2 A A A A A A 2 2 = + + 2 2 sec + 2 sec tan tan + x x x x 2 1 A 2 2 sec 2 sec tan tan x x x x 2 1 1 2 2 + 2 cos cos 2 A A 1 1 = + 2 2 2 sec 2 tan x x 2 2 = + + 2 2 1 ( 2 2+ = tan tan 4 ) 2 tan x x 2 x QED Back to Qs II Back to Qs II

  34. Summary You should be able to: Be able to derive all the trig identities Apply them when appropriate to solve trig equations Find exact values (addition formulas) Apply where appropriate to complete a Proof or show that question manipulate trig expressions fluently

  35. END

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