terms of powers (complete numbers)!

**lawochekel** · April 2nd 2012, 02:00 AM

express \cos^4\theta in terms of powers of \tan\theta and \cos\theta.

pls help on the above problem, thanks

**biffboy** · April 2nd 2012, 02:16 AM

I'll write x instead of theta
cos^4x=(1-sin^2x)^2=1-2sin^2x+sin^4x
But sinx=tanxcosx
So cos^4x=1-2tan^2xcos^2x+tan^4xcos^4x

**princeps** · April 2nd 2012, 02:19 AM

Originally Posted by lawochekel

express \cos^4\theta in terms of powers of \tan\theta and \cos\theta.

pls help on the above problem, thanks

$\tan^2 \theta = \frac{1-\cos^2 \theta}{\cos^2 \theta} \Rightarrow \cos^2 \theta = \frac{1}{1+\tan^2 \theta}$

Hence :

$\cos^4 \theta =\frac{\cos^2 \theta}{1+\tan^2 \theta}$

**Soroban** · April 2nd 2012, 04:41 AM

Hello, lawochekel!

A slightly different approach . . .

$\text{Express }\cos^4\theta\text{ in terms of powers of }\tan\theta\text{ and }\cos\theta.$

$\cos^4\!\theta \;=\;\cos^2\!\theta\cdot\cos^2\!\theta \;=\;\frac{\cos^2\!\theta}{\sec^2\!\theta} \;=\;\frac{\cos^2\!\theta}{\tan^2\!\theta + 1}$

**MathVideos** · April 2nd 2012, 05:03 AM

Or an even easier approach cos^4(theta) = cos^4(theta) + tan^0(theta) - 1

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