Question 6.6.4.6: Establishing an identity Establish the identity: sinθ/1 + co......
Establishing an identity
Establish the identity: {\frac{\sin\theta}{1+\cos\theta}}+{\frac{1+\cos\theta}{\sin\theta}}=2\csc\theta
The left side is more complicated. Start with it and proceed to add.
{\frac{\sin\theta}{1+\cos\theta}}+{\frac{1+\cos\theta}{\sin\theta}}={\frac{\sin^{2}\theta+(1+\cos\theta)^{2}}{(1+\cos\theta)\,(\sin\theta)}} Add the quotients.
={\frac{\sin^{2}\theta+1+2\cos\theta+\cos^{2}\theta}{(1+\cos\theta)\ (\sin\theta)}} Remove parentheses in the numerator.
={\frac{(\sin^{2}\theta+\cos^{2}\theta)\ +1\ +2\cos\theta}{(1\ +\cos\theta)\ (\sin\theta)}} Regroup.
={\frac{2+2\cos\theta}{(1+\cos\theta)\ (\sin\theta)}} Pythagorean Identity
=\frac{2\cancel{\left(1+\cos\theta\right)}}{\cancel{\left(1{+}\cos\theta\right)}\left(\sin\theta\right)} Factor and divide out.
={\frac{2}{\sin\theta}}=2\csc\theta Reciprocal Identity