How do you simplify the expression $(1+cosx)/sinx+sinx/(1+cosx)$ ?

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Answer 1 · 2016-08-30T18:35:02

The expression can be simplified to $2cscx$

Start by putting on a common denominator.

$=>((1 + cosx)(1 + cosx))/((sinx)(1 + cosx)) + (sinx(sinx))/((sinx)(1 + cosx))$

$=>(cos^2x + 2cosx + 1 + sin^2x)/(sinx(1 + cosx))$

Apply the pythagorean identity $cos^2x + sin^2x = 1$ :

$=>(1 + 2cosx + 1)/(sinx(1 + cosx)$

$=>(2 + 2cosx)/(sinx(1 + cosx))$

Factor out a $2$ in the numerator.

$=> (2(1 + cosx))/(sinx(1 + cosx))$

$=>2/sinx$

Finally, apply the reciprocal identity $1/sintheta = csctheta$ to get:

$=> 2cscx$

Hopefully this helps!

How do you simplify the expression (1+cosx)/sinx+sinx/(1+cosx)(1+cosx)/sinx+sinx/(1+cosx)?