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Solving a Quadratic Equation by Factorization Method

Tan A+cosec B...

Question

(TanA +cosec B)^2-(cot B-sec A)^2=2tanAcotB(cosecA+secB)

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Solution

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)

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Similar questions

Q. If A and B are complementary angles, then

(a) sin A = sin B
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(c) tan A = tan B
(d) sec A = cosec B

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