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Question

Mathematics Question on Trigonometric Functions

If m=tan⁡ θ+sin⁡ θm = \tan \, \theta + \sin \, \theta and n=tan⁡ θ−sin⁡ θn = \tan \, \theta - \sin \, \theta, then (m2−n2)2(m^2 - n^2)^2 is equal to

A

m nm \, n

B

4 m n4 \, m \, n

C

16 m n16 \, m \, n

D

4mn4 \sqrt{m n }

Answer

16 m n16 \, m \, n

Explanation

Solution

\left(m^{2} - n^{2}\right)^{2} = \left\\{\left(m+n\right)\left(m-n\right)\right\\}^{2} =\left\\{\left(2 \tan\theta\right)\left(2 \sin\theta\right)\right\\}^{2} =16tan⁡2θsin⁡2θ=16(tan⁡2θ−sin⁡2θ)= 16 \tan^{2} \theta \sin^{2} \theta= 16\left(\tan^{2} \theta - \sin^{2} \theta\right) =16mn= 16 mn (∵tan⁡2θsin⁡2θ=tan⁡2θ(1−cos⁡2θ)=tan⁡2θ−sin⁡2θ) \left(\because \tan^{2} \theta \sin^{2} \theta = \tan^{2} \theta\left(1 - \cos^{2} \theta\right) = \tan^{2} \theta - \sin^{2} \theta\right)