How is the step (red-boxed) formed?

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- Dec 7th 2010, 12:04 AMcloud5Intergration
How is the step (red-boxed) formed?

Attachment 20006 - Dec 7th 2010, 12:12 AMEducated
Using the trig identity $\displaystyle \sin^2 \theta + \cos^2 \theta = 1$

Then: $\displaystyle \sqrt{1-\sin^2 \theta} = \sqrt{\cos^2 \theta} = \cos \theta$

Since there is a $\displaystyle \cos \theta$ on both the denominator and numerator, they cancel out.

Then by using power reducing formulas:

$\displaystyle \sin^2 \theta = \dfrac{1-\cos(2\theta)}{2}$ - Dec 7th 2010, 12:15 AMDrSteve
$\displaystyle \sin^2 \theta = \frac{1-cos 2\theta}{2}$ is the standard power reducing formula which is worth memorizing for a calculus class since it is used so often.

You can derive it by writing $\displaystyle \cos 2\theta$ in terms of $\displaystyle \sin^2 \theta$ and solving for $\displaystyle \sin^2 \theta$.