Question

A student was asked to prove the trigonometric identity tangent of one half times x plus cotangent of one half times x equals 2 times cosecant x period Which of the following could be the first step in proving the identity? the quantity 1 minus cosine x end quantity over sine x plus sin x over the quantity 1 minus cosine x end fquantity equals 2 times cosecant x the quantity 1 minus cosine x end quantity over sine x plus the quantity 1 minus cosine x end quantity over sine x equals 2 times cosecant x sine x over the quantity 1 plus cosine x end quantity plus the quantity 1 minus cosine x end quantity over sine x equals 2 times cosecant xI onlyII onlyI and II onlyI, II, and III

Answer 1

Given:

$tan(\frac{1}{2}x)+cot(\frac{1}{2}x)=2cscx$

To start you use the formula for a half angle identity for tan.

$tan(\frac{1}{2}x)=\frac{1-cosx}{sinx}=\frac{sinx}{1+cosx}$

Cot is just the just 1/tan so we can just invert the half angle identity for tan.

$cot(\frac{1}{2}x)=\frac{sinx}{1-cosx}=\frac{1+cosx}{sinx}$

Therefore we can have a multitude of answers. But the only one available for the choices given is:

$\frac{1-cosx}{sinx}+\frac{sinx}{1-cosx}$

Answer:

1st one: I only