How can $\frac{1 + cos x}{sin x} + \frac{sin x}{1 + cos x} = 2 csc x$ $(1+cosx)/(sinx) + (sinx)/(1+cosx)=2cscx$ ?

Question

2607 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-05-04T07:07:25

We need

${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$

${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ $(a+b)^2=a^2+2ab+b^2$

$csc x = \frac{1}{sin x}$ $cscx=1/sinx$

Therefore,

$L H S = \frac{sin x}{cos x + 1} + \frac{cos x + 1}{sin x}$ $LHS=sinx/(cosx+1)+(cosx+1)/sinx$

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

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

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

$=2cancel(1+cosx)/(sinxcancel(cosx+1))$

$=2/sinx$

$=2cscx$

$=RHS$

$QED$

How can (1+cosx)/(sinx) + (sinx)/(1+cosx)=2cscx1+cosxsinx+sinx1+cosx=2cscx(1+cosx)/(sinx) + (sinx)/(1+cosx)=2cscx?