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?
$\endgroup$ 12 Answers
$\begingroup$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 $$\sin 2\left(\frac \theta 2\right) = \sin\left(2\cdot \frac \theta 2\right) = \sin \theta$$
$\endgroup$ $\begingroup$$$\cos^2 \left(\frac{\theta}{2}\right)− \sin^2 \left(\frac{\theta}{2}\right)=\cos \left(2\cdot\frac{\theta}{2}\right)=\cos\theta$$
$$2 \sin \left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)=\sin \left(2\cdot\frac{\theta}{2}\right)=\sin\theta$$
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