Question

How do you prove that `costheta-sinthetasin2theta=costhetacos2theta`?

Answer 1

`costheta-sinthetasin2theta=costheta-sintheta2sinthetacostheta=`

`=costheta-2sin^2thetacostheta=costheta(1-2sin^2theta)`

But we know that `cos2theta=cos^2theta-sin^2theta=1-sin^2theta-sin^2theta=1-2sin^2theta`

Then, we have `costheta-sinthetasin2theta=costhetacos2theta`

QED

Answer 2

`"see explanation"`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)sin2theta=2sinthetacostheta`

`•color(white)(x)cos2theta=1-2sin^2theta`

`"consider the left side"`

`costheta-sintheta(2sinthetacostheta)`

`=costheta-2sin^2thetacostheta`

`=costheta(1-2sin^2theta)`

`=costhetacos2theta`

`="right side "rArr"verified"`