1. \(\frac{cos A – sin A + 1}{cos A + sin A – 1} = cosec A + cot A\) using the identity \(cosec^2A = 1 + cot^2A\)
Proof:
\(LHS=\frac{cos A – sin A + 1}{cos A + sin A – 1}\)
\(=\)
\(=\)
\(=\)
\(= cosec A + cot A\)
\(=RHS\)
Hence proved.
2. \(\sqrt{\frac{1 + sin A}{1 – sin A}} = sec A + tan A\)
Proof:
LHS \(= \sqrt{\frac{1 + sin A}{1 – sin A}}\)
\(=\)
\(=\)
\(=\)
\(=\)
\(= sec A + tan A\)
\(=\) RHS
Hence proved.
Answer variants:
\(\frac{2cosec A(cosec A + cot A) – 2(cosec A + cot A)}{cot^2A – 1 – cosec^2A + 2cosec A}\)
\((cosec A + cot A)(2cosec A – 2 ) (2cosec A – 2)\)
\(\sqrt{\frac{(1 + sin A)(1 + sin A)}{(1 – sin A)(1 + sin A)}}\)
\(\sqrt{\frac{(1 + sin A)^2}{1 – sin^2A}}\)
\(\frac{2cosec^2A – 2cot A – 2cosec A + 2cot Acosec A}{cot^2A – (1 – cosec^2A + 2cosec A)}\)
\(\frac{1 + sin A}{cos A}\)
\(\sqrt{\frac{(1 + sin A)^2}{cos^2A}}\)