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Question
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Solution
In the given question, we need to prove `(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Here, we will first solve the LHS.
Now using `tan theta = sin theta/cos theta` and `cot theta = cos theta/sin theta` we get
`tan^2 A/(1 + tan^2 A) + cot^2 A/(1 + cot^2 A) = ((sin^2 A/cos^2 A))/((1 + sin^2 A/cos^2 A)) + ((cos^2 A/sin^2 A))/((1 + cos^2 A/sin^2 A))`
`= ((sin^2 A/cos^2 A))/(((cos^2 + sin^2 A)/cos^2 A)) + ((cos^2 A/sin^2 A))/(((sin^2 A + cos^2 A)/sin^2 A))`
`= ((sin^2 A/cos^2 A))/((1/cos^2 A)) + ((cos^2 A/sin^2 A))/((1/(sin^2 A)))` (using `sin^2 theta + cos^2 theta = 1`)
On further solving by taking the reciprocal of the denominator, we get,
`(sin^2 A/cos^2 A)/(1/cos^2 A) + (cos^2 A/sin^2 A)/(1/sin^2 A) = ((sin^2 A)/(cos^2 A)) (cos^2 A/1) + (cos^2 A/sin^2 A)(sin^2 A/1)`
`= sin^2 A + cos^2 A` (Using `sin^2 theta + cos^2 theta = 1`)
= 1
Hence proved.
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