Question

Prove that 1 / sec a - 1 +
1/ sec a + 1 = 2 cosec a cot a
​

Answer 1

$\sf\huge\red{\underbrace{ Question : }}$

Prove that :

$\sf \cfrac{1}{\sec\:a - 1} + \cfrac{1}{\sec\:a+1}=2\csc\:a\cot\:a$

$\sf\huge\orange{\underbrace{ Solution : }}$

★ LHS :

$\sf \implies \cfrac{1}{\sec\:a - 1} + \cfrac {1}{\sec\:a + 1}$

$\sf \implies \cfrac{(\sec\:a+ 1)+(\sec\:a - 1)}{(\sec\:a-1)(\sec\:a+1)}$

$\sf \implies \cfrac{\sec\:a+ 1+\sec\:a - 1}{(\sec\:a)^{2} - (1)^{2}}$

• (a + b)(a - b) = a² - b²

$\sf \implies \cfrac{2\sec\:a}{\sec^{2}\:a - 1}$

• sec² θ - 1= tan² θ

$\sf \implies \cfrac{2\sec\:a}{\tan^{2}\:a}$

$\sf \implies \cfrac{2 \times \frac{1}{\cos\:a}}{\frac{\sin^{2} \: a}{\cos^{2}\:a}}$

$\sf \implies \cfrac{2}{\cancel{\cos\:a}} \times \cfrac{\cancel{\cos^{2}\:a}}{\sin^{2}\:a}$

$\sf \implies \cfrac{2\cos\:a}{\sin^{2}\:a}$

• sin² θ = sin θ.sin θ

$\sf \implies \cfrac{2 \cos\:a}{\sin\:a\times \sin\:a}$

$\sf \implies 2 \times \cfrac{1}{\sin\:a} \times \cfrac{\cos\:a}{\sin\:a}$

• 1/sin θ = csc θ
• cos θ/sin θ = cot θ

$\sf \implies 2\csc\:a\cot\:a \: \: \tt\green{=RHS}$

$\blue{\bigstar}\underline{\boxed{\rm{\purple{\therefore Hence,\:it\:was\:proved.}}}}\:\blue{\bigstar}$