Question

prove that: √1+sinQ/1-sinQ + √1-sinQ/1+sinQ=2 sec Q

Answer 1

Answer

LHS = RHS

Given

$\bullet \;\; \rm \sqrt{\dfrac{1+sin\ \theta}{1-sin\theta}}+\sqrt{\dfrac{1-sin\theta}{1+sin\theta}}$

To Prove

$\rm \bullet \;\; 2\ sec\theta$

Proof

$\rm LHS\\\\\rightarrow \sqrt{\dfrac{1+sin\theta}{1-sin\theta}}+\sqrt{\dfrac{1-sin\theta}{1+sin\theta}}\\\\\rm Rationalize\ the\ denominator\\\\\rightarrow \sqrt{\dfrac{1+sin\theta}{1-sin\theta}\times \dfrac{1+sin\theta}{1+sin\theta}}+\sqrt{\dfrac{1-sin\theta}{1+sin\theta}\times \dfrac{1-sin\theta}{1-sin\theta}}\\\\\rightarrow \rm \sqrt{\dfrac{(1+sin\theta)^2}{1-sin^2\theta}}+\sqrt{\dfrac{(1-sin\theta)^2}{1-sin^2\theta}}$

$\rightarrow \rm \sqrt{\dfrac{(1+sin\theta)^2}{cos^2\theta}}+\sqrt{\dfrac{(1-sin\theta)^2}{cos^2\theta}}\\\\\bf Since\ ,sin^2\theta+cos^2\theta=1\\\\\rightarrow \rm \dfrac{1+sin\theta}{cos\theta}+\dfrac{1-sin\theta}{cos\theta}\\\\\rightarrow \rm \dfrac{1+sin\theta+1-sin\theta}{cos\theta}\\\\\rightarrow \rm \dfrac{2}{cos\theta}\\\\\rightarrow \rm 2\ sec\theta\\\\\rm RHS$