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

`1/(cosec^2A) = 1/(1+cot^2A)`

`sin^2A = 1/(1+cot^2A)`

`sinA = +- 1/(sqrt(1+cot^2A))`

`sqrt(1+cot^2A)` will always be positive as we are adding two positive quantities.

Therefore 

`sin A = 1/sqrt(1+cot^2A)`

we know that

⇒ `tan A =  (sin A)/(cos A)`

However  

⇒ `cot A = (cos A)/(sin A)`

Therefore, tan A = `1/cot A`

⇒ Also sec² A = 1 + tan² A

= `1+ 1/(cot^2A)`

= `(cot^2 A+1)/(cot^2 A)`

`sec A =sqrt(cot^2A+1)/cot A`