# Thread: Verify a trig identity

1. ## Verify a trig identity

Hello, I'm doing an induction problem and in the basis step I have to verify this identity (assuming $\displaystyle \sin{x/2}\neq 0$):

$\displaystyle \sin{x}=\frac{\sin{(2x)}}{4\sin^2{\left(\frac{x}{2 } \right) }}-\frac{2\cos{\left( \frac{3x}{2} \right) }}{2\sin{\left( \frac{x}{2} \right) }}$

Getting a common denominator, I get the RHS to be

$\displaystyle \frac{\sin{(2x)}-4\sin{\left( \frac{x}{2} \right) }\cos{\left( \frac{3x}{2} \right) }}{4\sin^2{\left( \frac{x}{2} \right) }}$

But where do I go from here? I'm afraid I have almost no background in trigonometry, and so don't know the strategy for these types of problems.

2. ## Re: Verify a trig identity

There may be an easier way, but if you convert the right hand side to be all in terms of (x/2) it works out pretty easily. Start with:

$\displaystyle \sin (2x) = \sin (\frac x 2 + \frac {3x} 2) = \sin (\frac x 2 ) \cos (\frac {3x} 2) + \cos (\frac x 2 ) \sin (\frac {3x} 2 )$

then apply the identities:

$\displaystyle \cos (3a) = \cos^3(a) - 3 \sin^2 (a) \cos(a)$

$\displaystyle \sin(3a) = 3 \cos^2(a) \sin (a) - \sin^3(a)$

using a = x/2.

You should end up with the right hand side being: $\displaystyle 2 \sin (\frac x 2) \cos ( \frac x 2)$, which is equivalent to $\displaystyle \sin (2 \times \frac x 2) = \sin(x)$

3. ## Re: Verify a trig identity

Thanks a ton!

4. ## Re: Verify a trig identity

You can also consider the attached solution.