# Chapter 5 - Section 5.1 - Proving Identities - 5.1 Problem Set - Page 279: 39

As left side transforms into right side, hence given identity- $\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$ is true.

#### Work Step by Step

Given identity is- $\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$ Taking L.S. $\cot\theta \cos \theta + \sin\theta$ = $\frac{\cos\theta}{\sin \theta} . \cos\theta + \sin\theta . \frac{\sin\theta}{\sin\theta}$ ( Using ratio identity for $\cot\theta$) = $\frac{\cos^{2}\theta}{\sin \theta} + \frac{\sin^{2}\theta}{\sin\theta}$ = $\frac{\cos^{2}\theta + \sin^{2}\theta }{\sin \theta}$ = $\frac{1 }{\sin \theta}$ ( Recall first Pythagorean identity, $\cos^{2}\theta + \sin^{2}\theta$ = 1) = $\csc\theta$ = R.S. As left side transforms into right side, hence given identity- $\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$ is true.

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