More practice questions

  1. Q5. Prove that 3 sin (π/6) sec (π/3) − 4 sin (5π/6) cot (π/4) = 1 (NCERT Example 10, page 64).

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    sin (5π/6) = sin (π − π/6) = 1/2, so the left side is 3 × (1/2) × 2 − 4 × (1/2) × 1 = 3 − 2 = 1.

  2. Q6. Prove that cos (π/4 + x) + cos (π/4 − x) = √2 cos x (NCERT Example 15, page 65).

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    By sum to product, the left side is 2 cos (π/4) cos x = 2 × (1/√2) × cos x = √2 cos x.

  3. Q7. Prove that sin 2x + 2 sin 4x + sin 6x = 4 cos2 x sin 4x (NCERT Exercise 3.3, Q14, where the first term is printed “sin2 x”; it must be read as sin 2x, since with sin2 x the statement fails at x = π/4).

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    sin 2x + sin 6x = 2 sin 4x cos 2x, so the left side is 2 sin 4x (cos 2x + 1) = 2 sin 4x × 2 cos2 x = 4 cos2 x sin 4x.

  4. Q8. Prove that cos 4x = 1 − 8 sin2 x cos2 x (NCERT Exercise 3.3, Q24).

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    cos 4x = 1 − 2 sin2 2x = 1 − 2(2 sin x cos x)2 = 1 − 8 sin2 x cos2 x.

  5. Q9. Prove that (cos 9x − cos 5x)/(sin 17x − sin 3x) = −sin 2x/cos 10x (NCERT Exercise 3.3, Q16).

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    cos 9x − cos 5x = −2 sin 7x sin 2x and sin 17x − sin 3x = 2 cos 10x sin 7x. Cancelling 2 sin 7x (where it is not 0) leaves −sin 2x/cos 10x.

  6. Q10. tan x = −4/3 with x in the second quadrant. Find sin (x/2), cos (x/2) and tan (x/2) (NCERT Miscellaneous Exercise, Q8).

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    cos x = −3/5 (second quadrant). π/2 < x < π gives π/4 < x/2 < π/2, so both half-angle values are positive. sin2 (x/2) = (1 − cos x)/2 = 4/5 and cos2 (x/2) = (1 + cos x)/2 = 1/5, so sin (x/2) = 2/√5, cos (x/2) = 1/√5 and tan (x/2) = 2.