Question

( tan a + cosec b ) - ( cot b - sec a )^2 = 2 tan a cot b ( cosec a^2 - sec b )

Answer 1

(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)

Answer 2

Answer:

Step-by-step explanation:

LHS

= (tanA + cosecB)^2 - (cotB-secA)^2

= (tan^2A + 2tanAcosecB + cosec^2B) - (cot^2B - 2cotBsecA + sec^2A)

= tan^2A + cosec^2B + 2tanAcosecB - cot^2B - sec^2A + 2cotBsecA

Substituting (tan^2A = sec^2A - 1) and (cosec^2B = 1 + cot^2B) in the above step:

(sec^2A - 1) + (1 + cot^2B) + 2tanAcosecB - cot^2B - sec^2A + 2cotBsecA

= 2tanAcosecB + 2cotBsecA

= 2(tanAcosecB + cotBsecA)

RHS

= 2tanAcotB(cosecA + secB)

= 2tanAcotB((1 / sinA) + (1 / cosB))

= 2(tanA / sinA)cotB + 2tanA(cot B / cosB)

= 2(sinA / (cosAsinB))cotB + 2tanA(cosB / (sinBcosB))

= 2(1 / cosA)cotB + 2 tanA(1 / sinB)

= 2secAcotB + 2tanAcosecB

= 2(secAcotB + tanAcosecB)