Question

Prove the following identity:
cos A /1+ sin A
+
1+ sin A/ cos A
= 2 sec A
​

Answer 1

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Answer 2

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Given:

$\to \sf \dfrac{cosA}{1 + sinA} + \dfrac{1 + sinA}{cosA} = 2secA$

Now take LHS:

$\to \sf \dfrac{cosA}{1 + sinA} + \dfrac{1 + sinA}{cosA}$

Take LCM

$\sf \to\dfrac{(cosA)(cosA) + (1 + sinA)(1 + sinA)}{cosA(1 +sin A)}$

$\sf \to\dfrac{(cos^{2} A)+ (1 + sinA) {}^{2} }{cosA(1 +sin A)}$

$\sf \to \dfrac{cos {}^{2} A + 1 + sin {}^{2}A + 2sin A}{cosA(1 +sinA) }$

$\sf \to \dfrac{1+ 1+ 2sin A}{cosA(1 +sinA) }$

$\sf \to \dfrac{2+ 2sin A}{cosA(1 +sinA) }$

$\sf \to \dfrac{2(1+ sin A)}{cosA(1 +sinA) }$

$\large\bf{\pink{\sf \to \dfrac{2 \cancel{(1+ sin A)}}{cosA \cancel{(1 +sinA)} }} }$

$\sf \to \dfrac{2}{cosA}$

$\sf \to {2}{secA}$

Hence Proved