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10.03 Angle sum and difference identities



Consider the statement $\cos90^\circ=\cos90^\circ+\cos0^\circ$cos90°=cos90°+cos0°.  Is this equation true?

If we evaluate each of the expressions, we get an equation that says $0=0+1$0=0+1. This statement is not true! What does that tell us about trigonometric functions?

In general, the distributive property does not apply to trigonometric functions. Instead, however, we can apply the sum and difference identities to evaluate sums and differences of angle measures.

Sum and Difference Identities

$\sin\left(A+B\right)=\sin A\cos B+\cos A\sin B$sin(A+B)=sinAcosB+cosAsinB

$\sin\left(A-B\right)=\sin A\cos B-\cos A\sin B$sin(AB)=sinAcosBcosAsinB


$\cos\left(A+B\right)=\cos A\cos B-\sin A\sin B$cos(A+B)=cosAcosBsinAsinB

$\cos\left(A-B\right)=\cos A\cos B+\sin A\sin B$cos(AB)=cosAcosB+sinAsinB


$\tan\left(A+B\right)=\frac{\tan A+\tan B}{1-\tan A\tan B}$tan(A+B)=tanA+tanB1tanAtanB

$\tan\left(A-B\right)=\frac{\tan A-\tan B}{1+\tan A\tan B}$tan(AB)=tanAtanB1+tanAtanB

These identities can be proven using the law of sines, which we will visit at a later time. For now, we can accept their truth and use them as shortcuts.

Worked example

Question 1

Evaluate: Find an exact value expression for $\sin15^\circ$sin15° 

Think: Notice that $15^\circ=45^\circ-30^\circ$15°=45°30°. Let's use $A=45^\circ$A=45° and $B=30^\circ$B=30° in the difference formula for the $\sin\left(A-B\right)$sin(AB).

Do: Substitute into the formula and evaluate the expressions.

$\sin15^\circ$sin15° $=$= $\sin\left(45^\circ-30^\circ\right)$sin(45°30°)

Substitute$15^\circ=45^\circ-30^\circ$15°=45°30° .

  $=$= $\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ$sin45°cos30°cos45°sin30°

Let $A=45^\circ$A=45° and $B=30^\circ$B=30° in the difference formula.

  $=$= $\frac{1}{\sqrt{2}}\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{1}{\sqrt{2}}$123212×12

Substitute the known exact values.

  $=$= $\frac{\sqrt{3}-1}{2\sqrt{2}}$3122


  $=$= $\frac{\sqrt{6}-\sqrt{2}}{4}$624

Rationalize the denominator.


Practice questions

Question 2

Given that $\sin A=\frac{24}{25}$sinA=2425 and $\cos B=\frac{20}{29}$cosB=2029, find the exact value of:

  1. $\sin\left(A+B\right)$sin(A+B)

  2. $\cos\left(A-B\right)$cos(AB)

Question 3

Using the expansion of $\cos\left(A+B\right)$cos(A+B), find the exact value of $\cos75^\circ$cos75°. Express the value in rationalized form.

Question 4

Find the exact value of $\tan\frac{\pi}{12}$tanπ12 in simplest radical form. Express the value in rationalized form.


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