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Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula.

(a) $\cos^2 \left( \cfrac{θ}{2} \right)− \sin^2 \left( \cfrac{θ}{2} \right)$

(b) $2 \sin \left( \cfrac{θ}{2} \right) \cos \left( \cfrac{θ}{2} \right) $

This is what I got: $$ (A) \ \cos 2 \left( \cfrac{θ}{2} \right) $$

$$(B) \ \sin 2 \left( \cfrac{θ}{2} \right) $$

However, the computer program I am using says that I am wrong. What is the right answer?

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  • $\begingroup$ You are entirely correct... although your computer program may be expecting you to write $\cos 2\frac\theta2$ as $\cos\theta$ $\endgroup$ – Mathmo123 Jul 20 '14 at 15:37
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You are correct, except you need to note that the factor of $2$ in $\cos 2\left(\frac\theta 2\right)$ multiplies the angle: $$\cos 2\left(\frac\theta 2\right)=\cos \left(2\cdot \frac \theta 2\right) = \cos \theta$$

Similarly $$\sin 2\left(\frac \theta 2\right) = \sin\left(2\cdot \frac \theta 2\right) = \sin \theta$$

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$$\cos^2 \left(\frac{\theta}{2}\right)− \sin^2 \left(\frac{\theta}{2}\right)=\cos \left(2\cdot\frac{\theta}{2}\right)=\cos\theta$$

$$2 \sin \left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)=\sin \left(2\cdot\frac{\theta}{2}\right)=\sin\theta$$

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