Question

1. \csc x + \cot x = \frac{\sin x}{1 - \cos x}

Explanation:

Step1: Express in terms of sine and cosine

Recall that \(\csc x=\frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\).
So, \(\csc x+\cot x=\frac{1 + \cos x}{\sin x}\)

Step2: Rationalize the right - hand side

Multiply the numerator and denominator of \(\frac{\sin x}{1-\cos x}\) by \(1 + \cos x\).

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Since \(1-\cos^{2}x=\sin^{2}x\) (by the Pythagorean identity \(\sin^{2}x+\cos^{2}x = 1\)), we have \(\frac{\sin x(1+\cos x)}{\sin^{2}x}=\frac{1+\cos x}{\sin x}\)

Answer:

The identity \(\csc x+\cot x=\frac{\sin x}{1 - \cos x}\) is verified.