Skill: Apply compound-angle identities

To apply a compound-angle identity, rewrite an angle as a sum or difference of two familiar angles, such as $30^\circ$, $45^\circ$, or $60^\circ$.

The main identities are

$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$ $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ $$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$

For differences, change the signs between the terms appropriately:

$$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ $$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$

Example: Find the exact value of $\sin 75^\circ$.

Step 1: Rewrite the angle as a sum of familiar angles.

$$75^\circ=45^\circ+30^\circ$$

So,

$$\sin75^\circ=\sin(45^\circ+30^\circ)$$

Step 2: Use the sine addition identity.

$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$

Therefore,

$$\sin(45^\circ+30^\circ) =\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$$

Step 3: Substitute the exact trigonometric values.

$$=\left(\frac{\sqrt2}{2}\right)\left(\frac{\sqrt3}{2}\right) +\left(\frac{\sqrt2}{2}\right)\left(\frac12\right)$$

Step 4: Simplify.

$$=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}$$

Thus,

$$\boxed{\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}}$$

Remember that trigonometric functions are not generally distributive over addition. For example,

$$\sin(A+B)\ne \sin A+\sin B.$$

You must use the appropriate compound-angle identity.