Question

How do you simplify `cotB + (sinB / (1+cosB))`?

Answer 1

`cscbeta`

Explanation:

`cotbeta`+`sinbeta/(1+cosbeta)`

1) Turn everything into sin and cos.

`cosbeta/sinbeta+sinbeta/(1+cosbeta)`

2) Use LCD and create a common numerator.

`(1+cosbeta)/(1+cosbeta)*cosbeta/sinbeta+sinbeta/(1+cosbeta)*sinbeta/sinbeta`

`((1+cosbeta)cosbeta+sin^2beta)/(sinbeta(1+cosbeta))`

3) Distibute `cosbeta` in the denominator.

`(cosbeta+(cos^2beta+sin^2beta))/(sinbeta(1+cosbeta))`

4) Pythagorean Identity in the denominator. Then cancel out matching denominator and numerator.

`cancel(cosbeta+(1))/(sinbeta (cancel(1+cosbeta)))`

5) Left with `1/sinbeta` = `cscbeta`

Answer 2

`csc(B)`

Explanation:

`cot(B)+frac{sin(B)}{1+cos(B)}=frac{cos(B)}{sin(B)}+frac{sin(B)}{1+cos(B)}frac{1-cos(B)}{1-cos(B)}`

`=frac{cos(B)}{sin(B)}+frac{sin(B)[1-cos(B)]}{1-cos^2(B)}`

`=frac{cos(B)}{sin(B)}+frac{sin(B)[1-cos(B)]}{sin^2(B)}`

`=frac{cos(B)}{sin(B)}+frac{1-cos(B)}{sin(B)}`

`=frac{1}{sin(B)}`

`=csc(B)`