Question

Pick 2 and prove the following Identities;
1.) sin x sec x = tan x
2.) cos^2 - sin^2 x = 2 cos^2 x
3.) tan x = sec x / csc x
4.) ( sin theta + cos theta )^2 = 1 + sin theta cos theta
5.) sin theta = cos theta / cot theta

Answer 1

1) LHS

$\sin(x) \times \sec(x) = \sin(x) \times \frac{1}{( \cos(x) ) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \ \: = \frac{ \sin(x) }{ \cos(x ) } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \tan(x)$
=RHS

2) LHS
${ \cos(x) }^{2} - { \sin(x) }^{2} \\ = { \cos(x) }^{2} - (1 - { \ \cos (x) }^{2} ) \\ = { \cos(x) }^{2} - 1 + { \ \cos(x) }^{2} \\ = 2 { \cos(x) }^{2} - 1$
=RHS {It will not only 2cos^2x}

hope it helps you....
THANK YOU