Question

# Prove that 1-(cos^2(x)/1+sin(x))

Trigonometric Functions
Prove that $$\displaystyle{1}-{\left(\frac{{{\cos}^{{2}}{\left({x}\right)}}}{{1}}+{\sin{{\left({x}\right)}}}\right)}$$

2021-05-22

We have to prove that $$\displaystyle{\left({1}-\frac{{{{\cos}^{{2}}{\left({x}\right)}}}}{{{1}+{\sin{{\left({x}\right)}}}}}\right)}={\sin{{\left({x}\right)}}}$$
Let us start from the left hand side. Note that $$\displaystyle{{\sin}^{{2}}{x}}+{{\cos}^{{2}}{x}}={1}$$. Then
$$(1-(\cos^2(x))=\frac{1+\sin(x)-\cos^2(x)}{1+(\sin(x))} =(\sin(x))+[1-\cos^2(x)]$$

$$=\frac{\sin(x)+\sin^2(x)}{1+(\sin(x))} =\frac{\sin(x)(1+\sin(x))}{1+\sin(x)} =\sin(x)$$
Hence the proof.