Sum and difference identities for sin, cos and tan: expansions, simplifications and exact values

What this dot point is asking

NESA wants you to know the sum and difference identities for sine, cosine and tangent, use them to expand or simplify expressions, and use them to compute exact values for non-standard angles like $15^\circ$ or $75^\circ$. These six formulas are the foundation of the whole trig-identities strand: the double-angle formulas, the half-angle formulas, the product-to-sum identities, the t-formula and the auxiliary-angle method are all consequences of them. The single feature that catches students is the sign flip in the cosine expansion (it goes the opposite way to the input sign), so that is the thing to lock down.

The answer

What "angle sum" means on the unit circle

The whole topic is about the angle $A + B$. On the unit circle, an angle $A$ reaches a point with coordinates $(\cos A, \sin A)$.

Adding a further angle $B$ swings the radius round to the point at angle $A + B$, whose coordinates are $\cos(A + B)$ and $\sin(A + B)$.

The sum and difference identities are exactly the formulas that express those coordinates of the combined angle in terms of the sines and cosines of $A$ and $B$ separately. That is why they let you reach a "new" angle ($75^\circ$) from angles you already know ($45^\circ$ and $30^\circ$).

The sum and difference identities

For sine,

For cosine, note the sign flips relative to the input,

For tangent,

The tangent denominator must be non-zero; if $1 \mp \tan A\tan B = 0$ the formula is undefined and you fall back on $\sin/\cos$.

Double-angle formulas: just set $A = B = \theta$

The double-angle formulas are not separate facts to memorise; they are the sum formulas with $B = A$:

The three forms of $\cos 2\theta$ come from substituting $\cos^2\theta + \sin^2\theta = 1$; pick whichever form suits the next step (the $1 - 2\sin^2\theta$ form is handy when you know $\sin\theta$). Setting both angles equal to $\theta$ is exactly the unit-circle picture of doubling an angle.

Half-angle formulas

Rearranging the $\cos 2\theta$ forms gives the power-reduction identities, and hence the half-angle formulas:

with the sign chosen from the quadrant of $\frac{\phi}{2}$. These give exact values such as $\cos 15^\circ$ and feed directly into the t-formula.

Exact values for non-standard angles

The standard angles $0, 30, 45, 60, 90^\circ$ have well-known exact values:

Writing a non-standard angle as a sum or difference of two standard angles, $15^\circ = 45^\circ - 30^\circ$, $75^\circ = 45^\circ + 30^\circ$, $105^\circ = 60^\circ + 45^\circ$, then applying the matching identity, produces its exact value.

How exam questions ask about sum and difference identities

• "Find the exact value of $\cos 75^\circ$ / $\sin 15^\circ$ / $\tan 105^\circ$": split into two standard angles, apply the right identity, simplify and rationalise.
• "Given $\sin\alpha$ and $\cos\beta$ (with quadrants), find $\sin(\alpha+\beta)$": get the missing ratios by Pythagoras with correct signs, then substitute into the sum identity.
• "Expand and simplify $\sin(\theta + \frac{\pi}{4}) + \sin(\theta - \frac{\pi}{4})$": expand both, cancel the opposite terms.
• "Prove the identity ...": expand each compound angle and collect; many proofs hinge on the cosine sign flip or on terms cancelling.
• "Find $\sin 3\theta$ / $\cos 3\theta$": write $3\theta = 2\theta + \theta$ and combine the sum identity with the double-angle formulas.
• "Hence solve / find the maximum ...": the identity is a stepping stone to an equation or to the auxiliary-angle form.

Edge cases worth knowing

• The tangent formula can be undefined. $\tan(45^\circ + 45^\circ)$ would need $1 - \tan 45^\circ\tan 45^\circ = 1 - 1 = 0$ in the denominator. The formula fails (as it must, since $\tan 90^\circ$ is undefined); reason via $\sin 90^\circ/\cos 90^\circ$ instead.
• Choose the efficient decomposition. $\sin 105^\circ = \sin(60^\circ + 45^\circ)$ is cleaner than $\sin(75^\circ + 30^\circ)$, because both $60^\circ$ and $45^\circ$ are standard while $75^\circ$ is not.
• The half-angle sign depends on the quadrant. $\cos\frac{\phi}{2} = \pm\sqrt{\frac{1+\cos\phi}{2}}$ takes the sign of cosine in the quadrant of $\frac{\phi}{2}$, not of $\phi$; decide it before writing the surd.