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Question

Mathematics Question on Trigonometric Identities

(sec⁡θ+tan⁡θ)(1−sin⁡θ)(\sec \theta + \tan \theta)(1 - \sin \theta) is equal to:

A

sec⁡θ\sec \theta

B

sin⁡θ\sin \theta

C

cosec⁡θ\cosec \theta

D

cos⁡θ\cos \theta

Answer

cos⁡θ\cos \theta

Explanation

Solution

Simplify the expression:
(sec⁡θ+tan⁡θ)(1−sin⁡θ)(\sec \theta + \tan \theta)(1 - \sin \theta)
Substitute sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}:
1+sin⁡θcos⁡θ(1−sin⁡θ)\frac{1 + \sin \theta}{\cos \theta}(1 - \sin \theta)
Simplify further using the identity:
(1+sin⁡θ)(1−sin⁡θ)cos⁡θ=1−sin⁡2θcos⁡θ\frac{(1 + \sin \theta)(1 - \sin \theta)}{\cos \theta} = \frac{1 - \sin^2 \theta}{\cos \theta}
=cos⁡2θcos⁡θ=cos⁡θ= \frac{\cos^2 \theta}{\cos \theta} = \cos \theta