Prove the following identity. Make sure to include all steps taken. (1+cos\theta )/(sin\theta)+(sin\theta)/(1+cos\theta)=2csc\theta

Question

Prove the following identity. Make sure to include all steps taken.
`(1+cos\theta )/(sin\theta)+(sin\theta)/(1+cos\theta)=2csc\theta`

Answer 1

See below.

Explanation:

`(1+cos(\theta))/(sin(\theta)) +(sin(\theta))/(1+cos(\theta))=2csc(\theta)`

`((1+cos(\theta))(1+cos(\theta)))/(sin(\theta)((1+cos(\theta))) +((sin(\theta))(sin(\theta)))/((1+cos(\theta))(sin(\theta)))=2csc(\theta)`

`((1+cos(\theta))(1+cos(\theta))+(sin(\theta))(sin(\theta)))/(sin(\theta)((1+cos(\theta)))=2csc(\theta)`

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

Recall the identities:

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

`(1+2cos(\theta)+1)/(sin(\theta)(1+cos(\theta))) =2csc(\theta)`

`(2+2cos(\theta))/(sin(\theta)(1+cos(\theta))=2csc(\theta)`

`(2(1+cos(\theta)))/(sin(\theta)(1+cos(\theta))) =2csc(\theta)`

`(2)/(sin(\theta)) =2csc(\theta)`

`2csc(\theta)=2csc(\theta)`