Question

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

Answer 1

$cose{c}^{2}A-co{t}^{2}A=1\Rightarrow cose{c}^{2}A=1+co{t}^{2}A\Rightarrow si{n}^{2}A=\left(\frac{1}{1+co{t}^{2}A}\right)\Rightarrow sinA=\pm \sqrt{\left(\frac{1}{1+co{t}^{2}A}\right)}Now,\Rightarrow si{n}^{2}A=\left(\frac{1}{1+co{t}^{2}A}\right)\Rightarrow 1-co{s}^{2}A=\left(\frac{1}{1+co{t}^{2}A}\right)\Rightarrow co{s}^{2}A=1-\left(\frac{1}{1+co{t}^{2}A}\right)\Rightarrow co{s}^{2}A=\left(\frac{co{t}^{2}A}{1+co{t}^{2}A}\right)\Rightarrow se{c}^{2}A=\left(\frac{1+co{t}^{2}A}{co{t}^{2}A}\right)=1+\left(\frac{1}{co{t}^{2}A}\right)\Rightarrow secA=\pm \sqrt{1+\left(\frac{1}{co{t}^{2}A}\right)}$
And, $tanA.cotA=1$
$\Rightarrow tanA=\left(\frac{1}{cotA}\right)$