Question

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

Answer 1

Answer:

Second Option and Fourth Option

Step-by-step explanation:

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

We know that:

$cot(\theta) = \frac{1}{tan(\theta)}$

Then:

$cot^2(\theta)cos(2\theta) = cos(2\theta)\frac{1}{tan^2(\theta)}\\\\cot^2(\theta)cos(2\theta) =\frac{cos(2\theta)}{tan^2(\theta)}$

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:

$cot^2(\theta)cos(2\theta) = \frac{cos^2(\theta)}{sin^2(\theta)}[1-2sin^2(\theta)]\\\\cot^2(\theta)cos(2\theta) = \frac{cos^2(\theta)[1-2sin^2(\theta)]}{sin^2(\theta)}\\\\cot^2(\theta)cos(2\theta) =cot^2(\theta) - 2\frac{cos^2(\theta)[sin^2(\theta)]}{sin^2(\theta)}\\\\cot^2(\theta)cos(2\theta) =cot^2(\theta) - 2cos^2(\theta)$