Express the trigonometric ratios $sin\ A, sec\ A$ and $tan\ A$ in terms of $cot\ A$.

To do:

We have to express the trigonometric ratios $sin\ A, sec\ A$ and $tan\ A$ in terms of $cot\ A$.

Solution:  

We know that,

$\operatorname{cosec^2}\ A - cot^2\ A = 1$

Therefore,

$\operatorname{cosec}^{2}\ A=1+\cot ^{2}\ A$

$\operatorname{cosec}\ A=\sqrt{1+\cot ^{2}\ A}$

$\sin\ A=\frac{1}{\sqrt{1+\cot ^{2}\ A}}$

$\sec ^{2} \mathrm{~A}-\tan ^{2} \mathrm{~A}=1$

$\Rightarrow \sec ^{2} \mathrm{~A}=1+\tan ^{2} \mathrm{~A}$

$=1+\frac{1}{\cot ^{2} \mathrm{~A}}$

$=\frac{\cot ^{2}+1}{\cot ^{2} \mathrm{~A}}$

$\Rightarrow \sec \mathrm{A}=\frac{\sqrt{\cot ^{2} \mathrm{~A}+1}}{\cot \mathrm{A}}$

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

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Updated on: 10-Oct-2022

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