How do you prove $((1+cosx) / sinx) + (sinx / (1 + cosx)) = 2 csc x$ ?

Question

28373 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-04-15T21:42:42

$LHS$

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

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

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

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

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

$=2/sinx$

$=2cscx$

$=RHS$

This is because:

$a/b+c/d = (ad+bc)/(bd)$

And also because:

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

[Source http://here.](https://useruploads.socratic.org/7acMCBSERuurNmOuVCJL_new%20quest%201.png)

How do you prove ((1+cosx) / sinx) + (sinx / (1 + cosx)) = 2 csc x((1+cosx) / sinx) + (sinx / (1 + cosx)) = 2 csc x?