"Prove that sin⁡A(1+cos⁡A)+(1+cos⁡A)sin⁡A=2csc⁡A" was answered by our experts

Ramsey

Answered question

2021-10-30

Prove that

\frac{\sin A}{(1 + \cos A)} + \frac{(1 + \cos A)}{\sin A} = 2 \csc A

liingliing8

Skilled2021-10-31Added 95 answers

We use Trigonometry identities

\frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = \frac{\sin^{2} A + {(1 + \cos A)}^{2}}{(1 + \cos A) (\sin A)}

= \frac{\sin^{2} A + 1^{2} + {(\cos A)}^{2} + 2 \cos A}{(1 + \cos A) \sin A}

\frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = \frac{\sin^{5} A + 1 + \cos^{5} A + 2 \cos A}{(1 + \cos A) \sin A}

= \frac{1 + 1 + 2 \cos A}{(1 + \cos A) \sin A}

= \frac{2 + 2 \cos A}{(1 + \cos A) (\sin A)}

\frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = \frac{2 (1 + c s o A)}{(1 + \cos A) (\sin A)} = \frac{2}{\sin A} = 2 \csc A

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?