How to convert sec^2(x)sin(2x) to 2tan(x)

Question

anonymous · Answer

$$\sec ^2 (x)\sin(2x) = 2\tan(x) $$

anonymous · Answer

I know that $$1+\tan^2(x) = \sec^2(x)$$

anonymous · Answer

So i have $$(1+\tan^2(x))(\sin(2x))$$

anonymous · Answer

then sin(2x) = 2sin(x)cos(x) so I have $$(1+\tan^2(x))(2\sin(x)\cos(x))$$

unklerhaukus · Answer

use this double angle formula $$\sin(2\theta)=2\sin(\theta)\cos(\theta)$$ like you have already,

but instead of doing that to $\sec^2$, use $\sec=1/\cos$

anonymous · Answer

Ahhhhhhh ok so I get 1/cos^2

unklerhaukus · Answer

yeah, and you can now cancel a cos

anonymous · Answer

... my damn brain... I thought that that would not work because it was to simple

anonymous · Answer

Thanks man.

unklerhaukus · Answer

thats alright, these thing are tricky,
i usually try to convert every term to sin(x) and cos(x) first

anonymous · Answer

sounds good, now how would I do $$\tan(x)2\cos(2x)$$

anonymous · Answer

$$\frac{ \sin(x) }{ \cos(x) }2(one of the three...)$$

unklerhaukus · Answer

$one with a \cos (x)  in it$

anonymous · Answer

cool man, got that now... goodness gracious me I hope this type of differentiation equation does not pop up in my exam... the orrigional equation was to find the first derivative of $$y = \frac{ 1 }{ 2 }\tan(x)\sin(2x)$$

unklerhaukus · Answer

the original equation?

unklerhaukus · Answer

just apply the double angle formula and convert the tan to sines and cosines