Exact Values with Compound Angles

For an exact value, choose standard angles that add or subtract to the required input. Keep fractions and surds; a rounded calculator value is only a check.

Use the same identities backwards when products already match a formula. The two individual angles need not be standard.

Find $\tan75^\circ$ and $\sin18^\circ\cos42^\circ+\cos18^\circ\sin42^\circ$ exactly.

Use $75^\circ=45^\circ+30^\circ$; multiplying numerator and denominator by $\sqrt3+1$ removes the surd denominator. The second expression is $\sin60^\circ=\sqrt3/2$.

If a question gives sine or cosine as a fraction, use $\sin^2A+\cos^2A=1$ to find the other magnitude. Use the quadrant to choose its sign before substitution.