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प्रश्न
Prove the following trigonometric identities:
`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
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उत्तर
We have to prove `(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
Consider the expression
L.H.S
`(1 + tan^2 theta)/(1 + cot^2 theta) = (1 + tan^2 theta)/(1 + 1/(tan^2 theta))`
= `(1 +tan^2 theta)/((tan^2 theta + 1)/tan^2 theta)`
`= tan^2 theta (1 + tan^2 theta)/(1 + tan^2 theta)`
`= tan^2 theta`
= R.H.S
Again, we have
L.H.S
`((1 - tan theta)/(1 - cot theta))^2 = ((1 - tan theta)/(1 - 1/(tan theta)))^2`
`= ((1 - tan theta)/((tan theta - 1)/tan theta))^2`
`=[(tantheta(1-tantheta))/-(1-tantheta)]^2`
`=(-tantheta)^2=tan^2theta`
= R.H.S
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