Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 658: 12

See the explanation below.

Work Step by Step

To verify the given identity, $\tan \theta +\cot \theta =\sec \theta \csc \theta$ Recall Trigonometric Identities, \begin{align} & {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\ & \tan \theta =\frac{\sin \theta }{\cos \theta } \\ & \cot \theta =\frac{\cos \theta }{\sin \theta } \\ \end{align} Use the above identities and solve the left side of the given expression, \begin{align} & \tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\ & =\frac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{\sin \theta \cos \theta } \\ & =\frac{1}{\sin \theta \cos \theta } \\ & =\frac{1}{\sin \theta }\left( \frac{1}{\cos \theta } \right) \end{align} Recall Reciprocal Identities, \begin{align} & \csc \theta =\frac{1}{\sin \theta } \\ & \sec \theta =\frac{1}{\cos \theta } \\ \end{align} Therefore, $\tan \theta +\cot \theta =\sec \theta \csc \theta$ Hence, it is proved that the given identity holds true.

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