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  4. If tan2 θ = 2 tan2 φ + 1, then cos 2 θ + sin2 φ is equal to

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Q. If $\tan^2 \, \theta = 2 \, \tan^2 \, \phi + 1,$ then $\cos \, 2 \theta + \sin^2 \, \phi $ is equal to

Trigonometric Functions

Solution:

$\cos 2 \theta + \sin^{2} \phi = \frac{1-\tan^{2} \theta}{1+\tan^{2}\theta} + \sin^{2} \phi $
$ = \frac{1-\left(2 \tan^{2} \phi + 1 \right)}{1+\left(2\tan^{2} \phi + 1 \right)} + \sin^{2} \phi $
$ = \frac{-\tan^{2}\phi}{1+\tan^{2} \phi} + \sin^{2} \phi = - \frac{\tan^{2} \phi}{\sec^{2} \phi} + \sin^{2} \phi = 0 $