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

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

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$$

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