Prove the identity 1 + cos x 1 - cos x - 1 - cos x 1 + cos x ≡ 4 sin x tan x

Given that, the trigonometric expression is LHS = 1 + cos x 1 - cos x - 1 - cos x 1 + cos x = 1 + cos x 2 - 1 - cos x 2 1 - cos x 1 + cos x = 1 + cos 2 x + 2 cos x - 1 - cos 2 x + 2 cos x 1 - cos 2 x = 4 cos x 1 - cos 2 x We know that, sin 2 x + cos 2 x = 1 ⇒ LHS = 4 cos x sin 2 x = 4 sin 2 x cos x We know that, tan x = sin x cos x = 4 sin x tan x = RHS Hence, proved.

Given that a = 1 - sin θ 2 cos θ , show that 1 a = 2 ( 1 + sin θ ) cos θ .

Given that, a = 1 - sin θ 2 cos θ ⇒ 1 a = 2 cos θ 1 - sin θ Multiplying the numerator and denominator with ( 1 + sin θ ) on RHS ⇒ 1 a = 2 cos θ 1 - sin θ × 1 + sin θ 1 + sin θ = 2 cos θ 1 + sin θ 1 - sin 2 θ We know that, sin 2 θ + cos 2 θ ≡ 1 ⇒ RHS = 2 cos θ ( 1 + sin θ ) cos 2 θ = 2 ( 1 + sin θ ) cos θ Hence, proved.

Given that a = 1 - sin θ 2 cos θ . Hence, find sin θ and cos θ in terms of a .

Given that, a = 1 - sin θ 2 cos θ ⇒ cos θ = 1 - sin θ 2 a ......(i) Now consider the reciprocal of a ⇒ 1 a = 2 cos θ 1 - sin θ Multiplying the numerator and denominator with ( 1 + sin θ ) on RHS ⇒ 1 a = 2 cos θ 1 - sin θ × 1 + sin θ 1 + sin θ = 2 cos θ 1 + sin θ 1 - sin 2 θ We know that, sin 2 θ + cos 2 θ ≡ 1 ⇒ RHS = 2 cos θ ( 1 + sin θ ) cos 2 θ ⇒ 1 a = 2 ( 1 + sin θ ) cos θ ⇒ cos θ = 2 a ( 1 + sin θ ) ......(ii) Equating (i) and (ii) 1 - sin θ 2 a = 2 a ( 1 + sin θ ) 1 - sin θ = 4 a 2 ( 1 + sin θ ) 1 - 4 a 2 = 4 a 2 sin θ + sin θ 1 - 4 a 2 = sin θ 4 a 2 + 1 ⇒ sin θ = 1 - 4 a 2 1 + 4 a 2 From (i): cos θ = 1 - sin θ 2 a ⇒ sin θ = 1 - 2 a cos θ .....(iii) From(ii): 1 a = 2 ( 1 + sin θ ) cos θ cos θ = 2 a + 2 a sin θ sin θ = cos θ - 2 a 2 a .....(iv) Equating (iii) and (iv) 1 - 2 a cos θ = cos θ - 2 a 2 a 2 a - 4 a 2 cos θ = cos θ - 2 a 4 a = cos θ 1 + 4 a 2 ∴ cos θ = 4 a 1 + 4 a 2 .

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Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 5 - Trigonometry>EXERCISE 5G

Sue Pemberton Solutions for Chapter: Trigonometry, Exercise 10: EXERCISE 5G

Author:Sue Pemberton

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton Solutions for Chapter: Trigonometry, Exercise 10: EXERCISE 5G

Attempt the practice questions on Chapter 5: Trigonometry, Exercise 10: EXERCISE 5G with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book solutions are prepared by Experienced Embibe Experts.