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प्रश्न
Prove the following trigonometric identities:
sin2 A cot2 A + cos2 A tan2 A = 1
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उत्तर
We have to prove `sin^2 A cot^2 A + cos^2 A tan^2 A = 1`
We know that `sin^2 A + cos^2 A = 1`
So,
`sin^2 A cot^2 A + cos^2 A tan^2 A = sin^2 A (cos^2 A)/(sin^2 A) + cos^2 A(sin^2 A)/(cos^2 A)`
`= cos^2 A + sin^2 A`
= 1
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Activity:
L.H.S. = `square`
= (sin2A + cos2A) `(square)`
= `1 (square)` ...`[sin^2"A" + square = 1]`
= `square` – cos2A ...[sin2A = 1 – cos2A]
= `square`
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Solution :
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= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
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∴ cotθ + tanθ = cosecθ × secθ
