Question

Prove the following trigonometric identities:

$\frac{1}{secA-1}+\frac{1}{secA+1}=2cosecAcotA$

Answer 1

Ans:

To prove,

1secA–1+1secA+1=2cosecAcotA1secA–1+1secA+1=2cosecAcotA

Considering left hand side (LHS),

= secA+1+secA−1(secA+1)(secA−1)secA+1+secA−1(secA+1)(secA−1)

= 2secA(sec2A−1)2secA(sec2A−1)

= 2secA(tan2A)2secA(tan2A)

= 2cos2A(cosAsin2A)2cos2A(cosAsin2A)

= 2cosA(sin2A)2cosA(sin2A)

= 2cosA(sinA)(sinA))2cosA(sinA)(sinA))

= 2cosec A cot A

Therefore, LHS = RHS

Hence proved