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Question
Prove the following trigonometric identities.
`cot^2 A cosec^2B - cot^2 B cosec^2 A = cot^2 A - cot^2 B`
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Solution
L.H.S = `cot^2 A cosec^2B - cot^2 B cosec^2 A`
`= cot^2 A(1+ cot^2 B) - cot^2 B(1 + cot^2 A)` (∵ `1 + cot^2 theta = cosec^2 theta`)
`= cot^2 A + cot^2 A cot^2 B - cot^2 B cot^2 A`
`= cot^2 A - cot^2 B`
Hence proved
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Activity:
L.H.S = `square`
= `square (1 - (sin^2theta)/(tan^2theta))`
= `tan^2theta (1 - square/((sin^2theta)/(cos^2theta)))`
= `tan^2theta (1 - (sin^2theta)/1 xx (cos^2theta)/square)`
= `tan^2theta (1 - square)`
= `tan^2theta xx square` .....[1 – cos2θ = sin2θ]
= R.H.S
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