Question

Evaluate 1/1-sin alpha + 1/1+ sin alpha

Answer 1

To solve :-

$\dfrac{1}{1 - \sin\alpha } + \dfrac{1}{1 + \sin\alpha }$

‎ ‎ ‎

Solution :-

$\dashrightarrow \dfrac{1}{1 - \sin\alpha } + \dfrac{1}{1 + \sin\alpha }$

Taking LCM

${ \dashrightarrow \dfrac{1 + \sin\alpha + 1 - \sin\alpha}{(1 - \sin\alpha) (1 + \sin\alpha)} }$

Now simplifying numerator

${ \dashrightarrow \dfrac{2}{(1 - \sin\alpha) (1 + \sin\alpha)} }$

Apply identity in denominator (a+b) (a-b) = a² - b² where a= 1 and b = sin α

${ \dashrightarrow \dfrac{2}{(1 - \sin ^{2} \alpha) } }$

Apply formula (1 - sin²α) = cos²α

${ \dashrightarrow \dfrac{2}{( \cos^{2} \alpha) } }$

Apply formula cos²α = 1/sec²α

${ \dashrightarrow 2 \sec^{2} \alpha }$

This is the required answer.