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