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

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

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We know that \csc^2A -\cot^2A = 1
(i)
\\\Rightarrow \frac{1}{\sin^2A}= 1+\cot^2A\\ \Rightarrow\sin^2A = \frac{1}{1+\cot^2A}\\ \Rightarrow \sin A = \frac{1}{\sqrt{1+\cot^2A}}

(ii) We know the identity of 
\\\sec^2A - \tan^2A = 1\\ \frac{1}{\cos^2A} = 1+\tan^2A=1+\frac{1}{\cot ^2A}\\ \frac{\cot^2A}{1+\cot^2A}=\cos^2A\\ \frac{\cot A}{\sqrt{1+\cotÂ}} = \cos A

(iii) \tan A = \frac{1}{\cot A}

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manish

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