Question

Which are simplified forms of the expression cot^2θcos2θ. Select two of the following there must be two selections!

Answer 1

Second Option and Fourth Option

Explanation:

`cot ^ 2(\theta) cos(2\theta)`

We know that:

`cot(\theta) = (1)/(tan(\theta))`

Then:

`cot^2(\theta)cos(2\theta) = cos(2\theta)(1)/(tan^2(\theta))\\\\cot^2(\theta)cos(2\theta) =(cos(2\theta))/(tan^2(\theta))`

Also:

For the sum of angles identity:

`cos(2\theta) = cos(\theta + \theta)\\\\cos(\theta + \theta) = cos(\theta)cos(\theta) - sin(\theta)sin(\theta)\\\\cos(2\theta) = cos^2(\theta) - sin^2(\theta)\\\\cos(2\theta) = (1-sin^2(\theta)) - sin^2(\theta)\\\\cos(2\theta) = 1-2sin^2(\theta)`

Then:

`cot^2(\theta)cos(2\theta) = (cos^2(\theta))/(sin^2(\theta))[1-2sin^2(\theta)]\\\\cot^2(\theta)cos(2\theta) = (cos^2(\theta)[1-2sin^2(\theta)])/(sin^2(\theta))\\\\cot^2(\theta)cos(2\theta) =cot^2(\theta) - 2(cos^2(\theta)[sin^2(\theta)])/(sin^2(\theta))\\\\cot^2(\theta)cos(2\theta) =cot^2(\theta) - 2cos^2(\theta)`