Question

Prove that sin A - cos A +1\sin A +cos A -1= 1\sec A - tan A, using the identity sec²A=1+tan²A.

Answer 1

Hi,

LHS = sinA-cosA+1/sinA+cosA-1

divide both numerator and denominator by cosA

LHS=(tanA−1+secA)/(tanA+1−secA)LHS=(tanA−1+secA)/(tanA+1−secA)

Now

sec2A=1+tan2Asec2A=1+tan2A

sec2A−tan2A=1sec2A−tan2A=1

Using above relation at denominator of LHS

LHS=(tanA−1+secA)/(tanA−secA+sec2A−tan2A)LHS=(tanA−1+secA)/(tanA−secA+sec2A−tan2A)

LHS=(tanA−1+secA)/((secA−tanA)(−1+secA+tanA))LHS=(tanA−1+secA)/((secA−tanA)(−1+secA+tanA))

LHS=1/(secA−tanA)LHS=1/(secA−tanA)

LHS=RHSLHS=RHS

Hence Proved.

I think above proof will clear your doubt,

All the best.