Question

`Evaluate: \sqrt{ \frac {1 - sin(x)}{1 + sin(x)​​}} \\`

Solve this problem please..​​

Answer 1

`\large\underline{\sf{Solution-}}`

We have to evaluate the given expression.

`\rm = \sqrt{ (1 - \sin(x) )/(1 + \sin(x) ) }`

If we multiple both numerator and denominator by 1 - sin(x), then the value remains same. Let's do that.

`\rm = \sqrt{ ([1 - \sin(x)][1 - \sin(x) ])/([1 + \sin(x)][1 - \sin(x) ]) }`

`\rm = \sqrt{ ([1 - \sin(x)]^(2))/(1- \sin^(2) (x) ) }`

We know that:

`\rm \longmapsto { \sin}^(2)(x) + \cos^(2)(x) = 1`

`\rm \longmapsto \cos^(2)(x) = 1 - { \sin}^(2)(x)`

Therefore, the expression becomes:

`\rm = \sqrt{ ([1 - \sin(x)]^(2))/(\cos^(2) (x))}`

`\rm = (1 - \sin(x))/(\cos(x))`

`\rm = (1)/(\cos(x)) - ( \sin(x) )/( \cos(x) )`

`\rm = \sec(x) - \tan(x)`

Answer 2

`\sqrt{ ( \cos(n) )/(1 + \: \sin(n) ) }`

see the attachment!!

hope it helps

#carryolearning