Just started learning about some Identities in class and quite haven't got the hang of it yet. Some Help for the following is much appreciated.
4sin(x/4)cos(x/4)cos(x/2)=sinx
tan2x+sec2x=cosx+sinx/cosx-sinx
Thanks
$\displaystyle \text{LHS} = 4 \sin \left(\frac{x}{4}\right) \cos \left(\frac{x}{4}\right) \cos \left(\frac{x}{2}\right)$
$\displaystyle \text{LHS} = 2 \cdot \underbrace{{\color{blue}2 \sin \left(\frac{x}{4}\right) \cos \left(\frac{x}{4}\right)}}_{\sin 2\theta = 2\sin \theta \cos \theta} \cos \left(\frac{x}{2}\right)$
Hopefully you'll see where to go from that.
For the second one, these double angle identities should help:
$\displaystyle \tan (2x) = \frac{2\tan x}{1 - \tan^{2} x}$$\displaystyle .\quad \text{and} \quad .$$\displaystyle \cos (2x) = \frac{1 - \tan^{2} x}{1 + \tan^{2}x}$
$\displaystyle \text{LHS} = \frac{2\tan x}{1 - \tan^{2} x} + \frac{1}{\cos 2x}$
$\displaystyle \text{LHS} = \frac{2\tan x}{1 - \tan^{2} x} + \frac{1 + \tan^{2} x}{1 - \tan^{2} x}$
Combine, factor, and convert into sin and cos
Maybe I should of clarified my original post. The problem I'm having is the algebra and how to combine these identities to simplify until I get cos & sin. I can Identify the identities and change them from double & Half-angle to the basic setup. I just struggle to simplify. Thanks for the Help o_O. Another push and maybe a few tips to help me get rolling would be great. I realize I do need more practice.
$\displaystyle \text{LHS} = 2 \cdot \underbrace{{\color{blue}2 \sin \left(\frac{x}{4}\right) \cos \left(\frac{x}{4}\right)}}_{\sin 2\theta = 2\sin \theta \cos \theta} \cos \left(\frac{x}{2}\right)$
Ok well, highlighted in blue looks like the right side of the identity: $\displaystyle \sin \2 \theta = 2 \sin \theta \cos \theta$ with $\displaystyle \theta = \frac{x}{4}$. So, if we convert it to the left side form, we get:
$\displaystyle 2 \cdot \underbrace{{\color{blue} \sin \left(2 \cdot \frac{x}{4}\right)}}_{\theta = \frac{x}{4}} \cos \left(\frac{x}{2}\right)$
$\displaystyle = 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)$
Doesn't this look familiar again?
-------------------------------------
As I suggested, combine the fractions:
$\displaystyle
\text{LHS} = \frac{2\tan x}{1 - \tan^{2} x} + \frac{1 + \tan^{2} x}{1 - \tan^{2} x}
$
$\displaystyle \text{LHS} = \frac{\tan^{2} x + 2 \tan x + 1}{1 - \tan^{2} x}$
Factor both top and bottom:
$\displaystyle \frac{\left(\tan x + 1\right){\color{blue}\left(\tan x + 1\right)}}{\left(1 - tan x\right){\color{blue}\left(1 + \tan x\right)}}$
Cancel and convert to sin and cos. See if you can carry on from here. Knowing how to manipulate fractions is good when coming to doing these identities. If you're struggling a bit, don't worry. It'll come with practice