# Which Trigonemetric Identity do I need to use?

• Aug 31st 2011, 11:22 AM
Srengam
Which Trigonemetric Identity do I need to use?
Hi all,

I'm stuck with just one step now of a long problem.

Basically, how do I get from

$\displaystyle (1+2sinx)(\frac{1}{2}-sinx)$

to

$\displaystyle 2cos(x)^{2}-\frac{3}{2}$

I've been told I need to use just one trigonometric identity, but I honestly can't think of any that will do it?

Many thanks.
• Aug 31st 2011, 11:28 AM
Siron
Re: Which Trigonemetric Identity do I need to use?
Expand:
$\displaystyle (1+2\sin(x))\cdot \left(\frac{1}{2}-\sin(x)\right)$

Afterwards use the fact $\displaystyle \sin^2(x)=1-\cos^2(x)$
• Aug 31st 2011, 11:40 AM
Srengam
Re: Which Trigonemetric Identity do I need to use?
Thanks but now theirs just one part I can't get after expanding,

how to get from

$\displaystyle (2sinx)(-sinx)$

to

$\displaystyle 2cos(x)^{2}-2$

this is the only missing step now,

many thanks again.
• Aug 31st 2011, 11:45 AM
Siron
Re: Which Trigonemetric Identity do I need to use?
If you expand:
$\displaystyle (1+2\sin(x))\cdot \left(\frac{1}{2}-\sin(x)\right)$ then you get:
$\displaystyle \frac{1}{2}+\sin(x)-\sin(x)-2\sin^2(x)=\frac{1}{2}-2\sin^2(x)$

Use $\displaystyle \sin^2(x)=1-\cos^2(x)$ so you get:
$\displaystyle =\frac{1}{2}-2(1-\cos^2(x))$
$\displaystyle =...$
• Aug 31st 2011, 11:48 AM
Srengam
Re: Which Trigonemetric Identity do I need to use?
Thank you so much,

I really appreciate it.
• Aug 31st 2011, 11:55 AM
Plato
Re: Which Trigonemetric Identity do I need to use?
Quote:

Originally Posted by Srengam
$\displaystyle (1+2sinx)(\frac{1}{2}-sinx)$
to
$\displaystyle 2cos(x)^{2}-\frac{3}{2}$

$\displaystyle (1+2sinx)(\frac{1}{2}-sinx)=\frac{1}{2}-2\sin^2(x)$
• Aug 31st 2011, 12:07 PM
Siron
Re: Which Trigonemetric Identity do I need to use?
Quote:

Originally Posted by Srengam
Thank you so much,

I really appreciate it.

You're welcome! :)