Question

if tan theta= cos2 alpha , prove that sin 2 theta =(1- tan4 alpha) / (1+tan4 alpha)

Answer 1

your answer for above question

Answer 2

given

$tan \theta = cos 2 \alpha$

We know

$sin2 \theta = \frac{2tan \theta}{1 + {tan}^{2} \theta } \\ \\ = \frac{2cos2 \alpha }{1 + {cos}^{2}2 \alpha } \\ \\ = \frac{2 \frac{1 - {tan}^{2} \alpha }{1 + {tan}^{2} \alpha } }{1 + \frac{(1 - {tan}^{2} \alpha) ^{2} }{(1 + {tan}^{2} \alpha ) ^{2} } } \\ \\ = \frac{2(1 - {tan}^{2} \alpha )(1 + {tan}^{2} \alpha ) }{( {1 + {tan}^{2} \alpha ) }^{2} +(1 - {tan}^{2} \alpha ) ^{2} } \\ \\ = \frac{2( 1 - {tan}^{4} \alpha ) }{2(1 + {tan}^{4} \alpha ) } \\ \\ = \frac{1 - {tan}^{4 } \alpha }{1 + {tan}^{4} \alpha }$