Solve the equation csc x - sin x = cot x cos x

Solve the equation csc x - sin x = cot x cos x

Question
Solve the equation \(\csc x - \sin x = \cot x \cos x\)

Answers (1)

2021-01-18
Work u\sing the left side
Use the reciptional identity: \(\csc x=\frac{1}{\sin x}\)
\(\csc x-\sin x=\frac{1}{\sin x-\sin x}\)
Simplify the right side into a \single expression:
\(\csc x-\sin x=(1-\sin^{2}x)-\sin x\)
Use the Pythagorean identity: \(\sin^{2}x+\cos^{2}x=1\)
\(\csc x-\sin x={\cos^{2}x}{\sin x}\)
Separate as:
\(\csc x-\sin x=\frac{\cos x}{\sin x}\times\cos x
Use the quotient identity: \(\cot x=\frac{\cos x}{\sin x}\)
\(\csc-\sin x=\cot x\cos x\)
0

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