1. Trigonometric Identities

Hello all, I have done a question but I just need clarification if it is correct:

Cot (x/2) = (1 + cos(x)) / sin(x)

what is the largest set of values of x for which this identity is valid?

i reasoned here that because they are equal to one another that either function can be used to solve for a value. this meant that for the identity to be invalid both functions would have to have a sin value of 0. this only happens at 2 * n * pi where n is any integer.

Am I correct?

2. Re: Trigonometric Identities

Well if the equation doesn't hold true for all values of $\displaystyle \displaystyle x$, then it's not an identity.

\displaystyle \displaystyle \begin{align*} \frac{1 + \cos{x}}{\sin{x}} &= \frac{1 + \cos{\left(2\cdot \frac{x}{2}\right)}}{\sin{\left(2\cdot \frac{x}{2}\right)}} \\ &= \frac{1 + \cos^2{\frac{x}{2}} - \sin^2{\frac{x}{2}}}{2\sin{\frac{x}{2}}\cos{\frac{ x}{2}}} \\ &= \frac{2\cos^2{\frac{x}{2}}}{2\sin{\frac{x}{2}}\cos {\frac{x}{2}}} \\ &= \frac{\cos{\frac{x}{2}}}{\sin{\frac{x}{2}}} \\ &= \cot{\frac{x}{2}}\end{align*}

It's true for all $\displaystyle \displaystyle x$.

3. Re: Trigonometric Identities

gah, I suck

Just a quick question though, since cot (x/2) = cos (x/2) / sin (x/2) how come you didnt use sin (x/2) as a restriction since it cannot ever be equal to zero?\

4. Re: Trigonometric Identities

Originally Posted by andrew2322
gah, I suck

Just a quick question though, since cot (x/2) = cos (x/2) / sin (x/2) how come you didnt use sin (x/2) as a restriction since it cannot ever be equal to zero?\

Well if the equation doesn't hold true for all values of $\displaystyle \displaystyle x$, then it's not an identity.