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

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Answer 1 · 2015-11-05T06:34:38

$cscbeta$

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

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 · 2015-11-05T06:38:30

$csc(B)$

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

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

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

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

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

$=csc(B)$

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