# Thread: terms of powers (complete numbers)!

1. ## terms of powers (complete numbers)!

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

pls help on the above problem, thanks

2. ## Re: terms of powers (complete numbers)!

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

3. ## Re: terms of powers (complete numbers)!

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 :

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

4. ## Re: terms of powers (complete numbers)!

Hello, lawochekel!

A slightly different approach . . .

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

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

5. ## Re: terms of powers (complete numbers)!

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