there are 2 proving identities problem that i cannot solve  1. CosecA-secA/cosecAsecA =cosA-sinA 2 sin^2A-tanA/cos^2A-cotA = sec^2A-1  Thankyou for helping me!!

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gsenviro | College Teacher | (Level 1) Educator Emeritus

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1.) Simplify the equation first,

Left hand side = (cosecA-secA)/cosecA secA = (cosecA/cosecAsecA) - (secA/cosecAsecA)

= 1/secA   - 1/cosecA

using the identity, sec A = 1/cosA and cosecA = 1/sinA, we get

LHS= 1/secA  - 1/cosecA = 1/(1/cosA)  - 1/(1/sinA) = cosA-sinA = RHS. Hence proved.

2) simplify everything in terms of sin and cos, using tanA = sinA/cosA and cotA = 1/tanA

LHS = (sin^2A-tanA)/(cos^2A-cotA) = (sin^2A - sinA/cosA)/(cos^2A - cosA/sinA) 

= [(sin^2A cosA-sinA)/cosA]/[(cos^2AsinA-cosA)/sinA] = [(sinA/cosA)x(sinAcosA-1)]/[(cosA/sinA)x((cosAsinA-1)] = (sinA/cosA)/(cosA/sinA)   ,after cancelling out (sinAcosA-1) from both numerator and denominator.

LHS = sin^2A/cos^2A = (1-cos^2A)/cos^2A = 1/cos^2A  - 1 = sec^2A -1= RHS

Hence proved.

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