Question

Someone help me with proving identities? and please show the work if you can!

Answer 1

See Below.

Explanation:

11)

We have:

`\displaystyle \cos\theta\cdot \cot\theta+\sin^2\theta\cdot \csc\theta=\csc\theta`

Rewrite:

`\displaystyle \cos\theta\left((\cos\theta)/(\sin\theta)\right)+\sin^2\theta\left((1)/(\sin\theta)}\right)=\csc\theta`

Multiply:

`\displaystyle (\cos^2\theta)/(\sin\theta)+(\sin^2\theta)/(\sin\theta)=\csc\theta`

Combine:

`\displaystyle (\cos^2\theta+\sin^2\theta)/(\sin\theta)=\csc\theta`

According to the Pythagorean Identity, sin²θ + cos²θ = 1. Hence:

`\displaystyle (1)/(\sin\theta)=\csc\theta`

Simplify:

`\csc\theta=\csc\theta`

12)

We have:

`\displaystyle (\cos^2\theta +\sin^2\theta)/(1+\tan^2\theta)=\cos^2\theta`

Pythagorean Identity:

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

From the Pythagorean Identity, if we divide both sides by cos²θ, we acquire that tan²θ + 1 = sec²θ. Hence:

`\displaystyle (1)/(\sec^2\theta)=\cos^2\theta`

Simplify:

`\cos^2\theta =\cos^2\theta`

13)

We have:

`\displaystyle (\csc\theta\cdot \cos\theta)/(\tan\theta +\cot\theta)=\cos^2\theta`

Rewrite:

`\displaystyle (1/\sin\theta\cdot \cos\theta)/(\sin\theta/\cos\theta+\cos\theta/\sin\theta)=\cos^2\theta`

Multiply both top and bottom by cosθsinθ. Hence:

`\displaystyle (\cos^2\theta)/(\sin^2\theta +\cos^2\theta)=\cos^2\theta`

By the Pythagorean Identity:

`\displaystyle (\cos^2\theta)/(1)=\cos^2\theta`

Simplify:

`\cos^2\theta=\cos^2\theta`