Question

Write all the other trigonometric ratios of $∠$ A in terms of sec A

Answer 1

We know that,
cos A = $\frac{1}{\text{sec A}}$
Also, sin2 A + cos2 A = 1
sin2 A = 1 - cos2 A

$⇒ \text{sin A} = \sqrt{1 - \left(\frac{1 }{ \text{sec A}}\right)^2}$

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

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

tan2 A + 1 = sec2 A
tan2 A = sec2 A - 1
tan A = $\sqrt{(\text{sec}^2 A - 1)}$
cot A =$\frac{ \text{cos A}}{\text{sin A}}$

$= \frac{\left(\frac{1}{\text{sec A}}\right) }{ \left[\frac{\sqrt{(\text{sec}^2 A - 1)}}{\text{sec A}}\right] }$
$= \frac{1 }{ \sqrt{(\text{sec}^2 A - 1)}}$

cosec A = $\frac{1}{\text{sin A}}$

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