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1.7.22 Proofs involving trigonometric functions (A-level only)

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Question 26
a.

Given that sin⁡θ≠±1\sin \theta \neq \pm 1sinθ=±1, prove the identity

11−sin⁡θ+11+sin⁡θ≡2sec⁡2θ \frac{1}{1 - \sin \theta} + \frac{1}{1 + \sin \theta} \equiv 2 \sec^2 \theta 1−sinθ1​+1+sinθ1​≡2sec2θ
[3]
b.

Hence, find the set of values of kkk for which the equation

11−sin⁡θ+11+sin⁡θ=k \frac{1}{1 - \sin \theta} + \frac{1}{1 + \sin \theta} = k 1−sinθ1​+1+sinθ1​=k

has real solutions. Fully justify your answer.

[3]
c.

Given that θ\thetaθ is in the second quadrant (reflex angle between 90∘90^\circ90∘ and 180∘180^\circ180∘) and

11−sin⁡θ+11+sin⁡θ=10 \frac{1}{1 - \sin \theta} + \frac{1}{1 + \sin \theta} = 10 1−sinθ1​+1+sinθ1​=10

find the exact value of tan⁡θ\tan \thetatanθ.

[3]

1.7.22 Proofs involving trigonometric functions (A-level only) Questions

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