Question

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

Answer 1

We know that,
cosec2 A = 1 + cot2 A
$\frac{1}{\text{cosec}^{2 }A} = \frac{1}{(1 + \text{cot}^2 A)}$
$\text{sin}^2 A = \frac{1}{(1 + \text{cot}^2 A)}$ ( $\text{As}\ \frac{1}{\text{cosec}^{2 }A} = \text{sin}^2 A$)

sin A =$± \frac{1}{\sqrt{(1 + \text{cot}^2 A)}}$

$\sqrt{(1 + \text{cot}^2 A)}$ will always be positive as we are adding two positive quantities.

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

We know that,
tan A = $\frac{\text{sin A} }{\text{ cos A}}$
However, we have,
cot A = $\frac{\text{cos A}}{\text{sin A}}$

Therefore, we have,
tan A =$\frac{ 1}{\text{cot A}}$

Also, sec2 A = 1 + tan2 A (Trigonometric Identity)
$= 1 + \frac{1}{\text{cot}^2 A}$

$= \frac{(\text{cot}^2 A + 1)} { \text{cot}^2 A}$

sec A =$\frac{\sqrt{(\text{cot}^2 A + 1)} }{\text{ cot A}}$