Question

Multiply and simplify: \(\frac{(\sin \theta+\cos \theta)(\sin \theta+\cos \theta)-1}{\sin \theta \cos \theta}\)

Answer 1

Use the square of a binomal:
\((a+b)^{2}=(a+b)(a+b)=a^{2}+2ab+b^{2}\)
\(=\frac{(\sin^{2}\theta+2\sin\cos \theta+\cos^{2}\theta)-1}{\sin \theta \cos \theta}\)
\(=\frac{(2\sin \theta \cos \theta+\sin^{2}\theta+\cos^{2}\theta)-1}{\sin \theta \cos \theta}\)
Use the Pythagorean identity: \(\sin^{2}\theta+\cos^{2}\theta=1\)
\(=\frac{(2\sin \theta\cos \theta+1)-1}{\sin \theta \cos \theta}\)
Simplify:
\(\frac{2\sin \theta \cos \theta}{\sin \theta \cos \theta}=2\)