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प्रश्न
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
उत्तर
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
L.H.S. = `(1 - tan^2theta)/(cot^2 theta - 1)`
= `1 - tan^2theta ÷ 1/(tan^2theta) - 1`
= `1 - tan^2theta ÷ (1 - tan^2theta)/(tan^2theta)`
= `(1 - tan^2theta) xx (tan^2theta)/((1 - tan^2 theta))`
= tan2 θ
L.H.S. = R.H.S.
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Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
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