# How do you prove (1+cosx)/sinx + sinx/(1+cosx) = 4?

Feb 23, 2016

$\frac{1 + \cos x}{\sin} x + \sin \frac{x}{1 + \cos x} = 2 \csc x$

(not $4$ as stated in the question)

#### Explanation:

$\frac{1 + \cos x}{\sin} x + \sin \frac{x}{1 + \cos x}$

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

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

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

=(2+2cosx)/(sinx(1+cosx) as ${\sin}^{2} x + {\cos}^{2} x = 1$

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

= $\frac{2}{\sin} x$ = $2 \csc x$

Feb 23, 2016

The equation has solutions x = 2 arc tan ( 2 $\pm$ sqrt3)

#### Explanation:

Converting to functions of x/2, the equation reduces to a quadratic in tan x/2. The roots are tan x/2 = 2 $\pm$ sqrt3