Find the exact value of $\cos (\alpha +\beta )$.

Find the exact value of $\sin (\alpha +\beta )$.

Show that $\theta =\arctan \frac{3}{x}-\arctan \frac{1}{x}$

Express $\theta$ as $\theta =\arctan \frac{ax}{x^2+b}$, where $a,b\in R$

Find the greatest possible value of the angle $\theta$, and the value of $x$ at which it occurs.

Find where the observer should stand so that the viewing angle is $25^{\circ }$.

Find all possible values of $\sin \theta$.

If $P$ lies in the third quadrant, find the exact values of $\sin (2\theta )$, $\cos (2\theta )$, and $\tan (2\theta )$.

Verify that the identity $1+{\tan }^2(2\theta )={\sec }^2(2\theta )$ holds for your values.

Let $Q=(\cos \theta -\sin \theta {)}^2$. Determine the maximum and minimum possible values of $Q$, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le \theta <2\pi$) for which the quantity $Q$ is greater than $1$.

Use the compound angle formula $\sin (A+B)=\sin A\cos B+\cos A\sin B$ to find the exact value of $\sin 75^{\circ }$.

Find the exact values of:

(i) $\cos 75^{\circ }$;

(ii) $\tan 75^{\circ }$.

Show that $\sin 75^{\circ }+\cos 75^{\circ }=\frac{\sqrt{6}}{2}$.

Solve $2\sin (x+60^{\circ })=\sqrt{3}$ for $0\le x<360^{\circ }$.

Using the formula for $\cos (A+B)$ and suitable double angle identities, prove the following identity: $\cos 4\theta =1-8{\sin }^2\theta {\cos }^2\theta$.

Find an expression for $\cos 6\theta$ in terms of $\sin \theta$ and $\cos \theta$ only.

Hence find $\frac{\cos 6\theta }{\cos 2\theta }$ in terms of $\sin \theta$ only.