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

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Solution

$c o s e c^{2} A - c o t^{2} A = 1 \Rightarrow c o s e c^{2} A = 1 + c o t^{2} A \Rightarrow s i n^{2} A = (\frac{1}{1 + c o t^{2} A}) \Rightarrow s i n A = \pm \sqrt{(\frac{1}{1 + c o t^{2} A})} N o w, \Rightarrow s i n^{2} A = (\frac{1}{1 + c o t^{2} A}) \Rightarrow 1 - c o s^{2} A = (\frac{1}{1 + c o t^{2} A}) \Rightarrow c o s^{2} A = 1 - (\frac{1}{1 + c o t^{2} A}) \Rightarrow c o s^{2} A = (\frac{c o t^{2} A}{1 + c o t^{2} A}) \Rightarrow s e c^{2} A = (\frac{1 + c o t^{2} A}{c o t^{2} A}) = 1 + (\frac{1}{c o t^{2} A}) \Rightarrow s e c A = \pm \sqrt{1 + (\frac{1}{c o t^{2} A})}$
And, $t a n A . c o t A = 1$
$\Rightarrow t a n A = (\frac{1}{c o t A})$

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