Write all the other trigonometric ratios of $\angle A$ in terms of $sec\ A$.

To do:

We have to write all the other trigonometric ratios of $\angle A$ in terms of $sec\ A$.

Solution:  

We know that,

$sin^2\ A + cos^2\ A = 1$

Therefore,

$\sin ^{2} A=1-\cos ^{2} A$

$=1-\frac{1}{\sec ^{2} A}$

$=\frac{\sec ^{2} A-1}{\sec ^{2} A}$

$\Rightarrow \sin A=\frac{\sqrt{\sec ^{2} A-1}}{\sec A}$

$\cos A=\frac{1}{\sec A}$

$\tan A=\frac{\sin A}{\cos A}$

$=\frac{\frac{\sqrt{\sec ^{2} A-1}}{\sec A}}{\frac{1}{\sec A}}$

$=\sqrt{\sec ^{2} A-1}$

$\operatorname{cosec} A=\frac{1}{\sin A}$

$=\frac{1}{\frac{\sqrt{\sec ^{2} A-1}}{\sec A}}$

$=\frac{\sec A}{\sqrt{\sec ^{2} A-1}}$

$\cot A=\frac{1}{\tan A}$

$=\frac{1}{\sqrt{\sec ^{2} A-1}}$

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Updated on: 10-Oct-2022

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