Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
We know that,
$\operatorname{cosec}^{2} A=1+\cot ^{2} A$
$\frac{1}{\operatorname{cosec}^{2} A}=\frac{1}{1+\cot ^{2} A}$
$\sin ^{2} A=\frac{1}{1+\cot ^{2} A}$
$\sin A=\pm \frac{1}{\sqrt{1+\cot ^{2} A}}$
$\sqrt{1+\cot ^{2} A}$ will always be positive as we are adding two positive quantities.
Therefore, $\sin A =\frac{1}{\sqrt{1+\cot ^{2} A }}$
We know that, $\tan A =\frac{\sin A }{\cos A }$
However, $\cot A=\frac{\cos A}{\sin A}$
Therefore, $\tan A =\frac{1}{\cot A }$
Also, $\sec ^{2} A=1+\tan ^{2} A$
$=1+\frac{1}{\cot ^{2} A}$
$=\frac{\cot ^{2} A+1}{\cot ^{2} A}$
$\sec A=\frac{\sqrt{\cot ^{2} A+1}}{\cot A}$
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