Question

Prove that (1+cotQ-cosecQ)(1+tanQ+secQ)=2​

Answer 1

Given

• $\bf(1 + cot \theta - cosec \theta)(1 + tan \theta + sec \theta) = 2$

To Prove

• LHS=RHS

Proof

LHS

$\bf \: (1 + cot \theta - cosec \theta)(1 + tan \theta + sec \theta) \\ \\ \implies \bf (1 + \dfrac{cos \theta}{sin \theta} - \dfrac{1}{sin \theta} )(1 + \dfrac{sin \theta}{cos \theta} + \dfrac{1}{cos \theta} ) \\ \\ \implies \: ( \bf\frac{sin \theta + cos \theta - 1}{sin \theta} )( \dfrac{cos \theta + sin \theta + 1}{cos \theta} ) \: \: \\ \\ \implies \bf \{ \frac{ {(sin \theta + cos \theta )}^{2} - 1}{sin \theta cos \theta} \} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \: \bf\frac{ {sin}^{2} \theta + {cos}^{2} \theta + 2sin \theta cos \theta }{sin \theta cos \theta} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \bf \frac{1 + 2sin \theta cos \theta}{sin \theta cos \theta} \implies \frac{1}{sin \theta cos \theta} + \frac{2sin \theta cos \theta}{sin \theta cos \theta} \\ \\ \implies \underline{ \boxed{ \huge \bf \: 2 = RHS}} \bf \: hence \: proved$