Question

I is known that cos2theta=cos ^2 theta-sin^2theta and that sin^2theta+cos^2 theta=1. Find the value of sin theta and cos theta in terms of cos2theta.

Answer 1

$cos2thetha + {sin}^{2} thetha \: = {cos}^{2} thetha$
on putting this value in another equation we get

$cos2thetha \: + {sin}^{2} thetha \: + {sin}^{2} thetha = 1$
cos2thetha +2sin^2 theta =1
sin thetha =
$\sqrt{ \frac{1 - cos2thetha}{2} }$
similarly it is of cos thetha

Answer 2

$\cos(2x) = { \cos }^{2} x - { \sin }^{2} x$
${ \sin }^{2} x + { \cos }^{2} x = 1$
Use the second identity in the first,

$\cos(2x) = { \cos }^{2} x - {(1 - { \ \cos }^{2}x) } \\ \cos(2x) =2 \cos ^{2} x - 1 \\ \frac{ \cos(2x ) + 1 }{2} = { \cos }^{2}x$

Also,

$\cos(2x) = { \cos }^{2} x - { \sin }^{2} x \\ \cos(2x) = 1 - { \sin }^{2} x - { \sin }^{2} x \\ \cos(2x) = 1 - 2 \sin^{2} x \\ \frac{1 - \cos(2x) }{2} = { \sin }^{2} x$

I have left it to the squared expressions. If you want in sin(x) and cos(x), take plus minus square roots on the left hand side.