Question

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

Trigonometric Functions
ANSWERED
asked 2021-05-21
Prove that \(\displaystyle{1}-{\left(\frac{{{\cos}^{{2}}{\left({x}\right)}}}{{1}}+{\sin{{\left({x}\right)}}}\right)}\)

Answers (1)

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.

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