Question

prove that tan²ø+cot²ø+2=sec²ø.cosec²ø​

Answer 1

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Here's your answer⤵⤵⤵

(tanA + cosecB)² - (cotB - secA)²

= tan²A + cosec²B + 2 tanA cosecB -(cot²B + sec²A - 2 cotB secA) ( using (a±b)² = a² + b² ± 2ab)

= tan²A + cosec²B + 2tanA cosecB - cot²B - sec²A + 2cotB secA

Rearranging above equation we get

cosec²B - cot²B - sec²A + tan²A + 2tanA cosecB + 2cotB secA

= (cosec²B - cot²B) - (sec²A - tan²A) + 2tanA cotB ( cosecB/cotB + secA/tanA )

= 1 - 1 + 2tanA cotB { (1/sinB)/(cosB/sinB) + (1/cosA)/(sinA/cosA) } ( As cosec²ø - cot²ø = 1 sec²ø - tan²ø = 1)

= 0 + 2tanA cotB { (1/sinB) x (sinB/cosB) + (1/cos A) x ( cosA/sinA) }

= 2tanA cotB ( 1/cosB + 1/sinA) = 2tanA cotB (secB + cosecA)

= 2tanA cotB (cosecA + secB)

Hope it helps you✌✌✌