cot(2(theta)) = (1/2)(cot(theta)-tan(theta))

[codycenters]

I need help solving this problem step-by-step.
cot(2(theta)) = (1/2)(cot(theta)-tan(theta))

 

[galactus]

Solve?. Are you sure you are supposed to solve?.

Looks like an identity to verify to me.

\(\displaystyle cot(2\theta)=\frac{1}{2}\left(cot(\theta)-tan(\theta)\right)\)

If so, write everything in terms of sin and cos.

Note the identity \(\displaystyle cos(2\theta)=cos^{2}\theta-sin^{2}\theta\)

 

[codycenters]

your right. it is suppose to be establishing the identity.
could you explain this in a little more detail please?

 

[codycenters]

i ended up getting (1/2) (sin^2 (theta)- cos^2 (theta))
but i believe the problem should end being cot^2(theta) - 1 / 2cot (theta)
how do i get to this step?

 

[soroban]

Hello, codycenters!

\(\displaystyle \text{Prove: }\:\cot 2\theta \:=\:\tfrac{1}{2}(\cot\theta - \tan\theta)\)

\(\displaystyle \text{We have: }\:\cot2\theta \;=\;\frac{1}{\tan2\theta} \;=\;\frac{1-\tan^2\theta}{2\tan\theta} \;=\;\tfrac{1}{2}\left(\frac{1}{\tan\theta} - \dfrac{\tan^2\theta}{\tan\theta}\right) \;=\;\tfrac{1}{2}(\cot\theta - \tan\theta)\)