Compound Angles: Build New Values from Known Ones

Derive the cosine of a difference

A compound angle is an angle written as a sum or difference, such as $\alpha+\beta$ or $\alpha-\beta$. Its trigonometric value is generally not the sum or difference of the separate values. For example, $\cos(60^\circ-30^\circ)=\sqrt3/2$, whereas $\cos60^\circ-\cos30^\circ=(1-\sqrt3)/2$.

To build the correct relationship, place points $A=(\cos\alpha,\sin\alpha)$ and $B=(\cos\beta,\sin\beta)$ on the unit circle. The distance formula gives

Now rotate both rays through $-\beta$. Rotation preserves their chord length. One point becomes $(1,0)$; the other has coordinates $(\cos(\alpha-\beta),\sin(\alpha-\beta))$. Computing the same squared distance in this position gives $2-2\cos(\alpha-\beta)$. Equate the two distance expressions to obtain

The derivation uses only circle coordinates, the distance formula and the fact that rotation preserves distances. It does not assume the addition identity it is trying to prove. The unit circle also makes the result valid beyond acute angles.

Replace $\beta$ by $-\beta$, using cosine’s evenness and sine’s oddness. The difference formula then becomes the sum formula:

Read the signs carefully: cosine of a difference has a plus between the products; cosine of a sum has a minus. Move the two angles in the construction and compare the resulting chord and coordinates.

Let the sine formula follow from cosine

Sine and cosine exchange roles under a complementary angle: $\sin t=\cos(\pi/2-t)$. Combine that fact with the cosine difference formula rather than treating another pair of identities as unrelated facts to memorise.

Write

Apply $\cos(U-V)=\cos U\cos V+\sin U\sin V$, with $U=\pi/2-\alpha$ and $V=\beta$. Since $\cos(\pi/2-\alpha)=\sin\alpha$ and $\sin(\pi/2-\alpha)=\cos\alpha$, this becomes

Now replace $\beta$ by $-\beta$. Cosine stays unchanged and sine changes sign:

Sine therefore keeps the sum/difference sign between its products. In each product, one factor is a sine and the other a cosine. The arguments are paired in opposite orders. Saying these features aloud is a useful way to check a line before substituting values.

All four sine/cosine sum and difference identities are valid for every real pair of angles. You do not need to assume both angles are acute. For a quadrant problem, however, the individual sine and cosine values must carry their correct signs. A known magnitude such as 3/5 is not a signed coordinate until the quadrant is specified.

Use exact forms when the angles are familiar. For unfamiliar inputs, a decimal check can reveal a sign error, but the identity itself is an exact equality. Keep both arguments in the same angle unit throughout a calculation.

Derive tangent and cotangent formulas with their conditions

Start with $\tan(\alpha+\beta)=\sin(\alpha+\beta)/\cos(\alpha+\beta)$. Substitute the two compound-angle formulas. If $\cos\alpha\ne0$ and $\cos\beta\ne0$, divide numerator and denominator by $\cos\alpha\cos\beta$. This gives

The fraction also needs a nonzero final denominator. Equivalently, the individual tangents and the tangent of the resulting angle must all be defined. A formula involving $\tan\alpha$ cannot be used when $\alpha=\pi/2$, even if $\tan(\alpha-\beta)$ happens to exist. In that situation return to sine and cosine or another valid angle split.

For cotangent, start with cosine divided by sine and divide the expanded terms by $\sin\alpha\sin\beta$. Where the individual and resulting cotangents are defined,

The order of the denominator in the difference formula matters. It follows directly from $\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$, so it is $\cot\beta-\cot\alpha$, not the reverse.

Conditions are part of the identity rather than a separate formality. They tell you whether the chosen algebraic route is permitted. Before plugging in values, check the two starting quotients and the final denominator. Then simplify exactly; rationalising a denominator can make the result easier to compare with another equivalent answer.

Double an angle by setting the two inputs equal

A double-angle formula is a compound-angle identity with the same angle used twice. Setting $\alpha=\beta=x$ in the sine sum formula gives

Doing the same in the cosine sum formula gives $\cos2x=\cos^2x-\sin^2x$. Replace either square using $\sin^2x+\cos^2x=1$ to obtain three equivalent forms:

These forms serve different purposes. If a problem supplies cosine, $2\cos^2x-1$ may be quickest. If it supplies sine, choose $1-2\sin^2x$. When a calculation contains a difference of squares, the first form may expose the structure immediately.

The tangent sum identity similarly gives

provided $\tan x$ and $\tan2x$ are defined. In this fraction, $1-\tan^2x\ne0$. The sine/cosine forms themselves remain valid for every real $x$.

Rearrange the cosine identities to reduce squares:

Notice that $\sin2x$, $2\sin x$ and $\sin^2x$ are three different instructions. The first doubles the angle, the second doubles the output and the third squares the output. A quick exact-angle test, such as $x=30^\circ$, can prevent them from being confused.

When roots are introduced later, remember that $\sqrt{\sin^2x}=|\sin x|$. A square-root step must not silently discard the sign information that the circle supplies.

Build triple angles from a double and a single angle

To calculate a triple angle, split $3x$ as $2x+x$. The sine sum identity and double-angle formulas give

As a quick check, put $x=0$: both sides of the sine identity are 0, while both sides of the cosine identity are 1. In the last step, replace $\cos^2x$ by $1-\sin^2x$ and collect the sine terms. Deriving the result this way explains both the coefficient 3 and the minus sign before the cubic term.

For cosine, use $\cos(2x+x)=\cos2x\cos x-\sin2x\sin x$. Replace $\cos2x$ by $2\cos^2x-1$, $\sin2x$ by $2\sin x\cos x$, and $\sin^2x$ by $1-\cos^2x$. Collecting terms yields

These two identities hold for every real angle. They also show why $\sin3x$ is not $3\sin x$: the cubic term is essential.

If $t=\tan x$, dividing the triple-angle sine and cosine expressions gives

This version requires $\cos x\ne0$ and $1-3t^2\ne0$, so that both $\tan x$ and $\tan3x$ exist. Keep the sine/cosine derivation available when a tangent expression is undefined; do not repair a zero denominator by cancellation without checking the original expression.

For a numerical problem, choose the formula that matches the ratio actually given. If cosine alone is supplied, the cosine cubic avoids an unnecessary square root and the sign choice it would require.

Turn a sum into a product

Sometimes the useful feature is a pair of different angles added together. Introduce their average and half-difference:

Now apply the compound-angle formulas to $\sin(u+v)+\sin(u-v)$. The $\cos u\sin v$ terms cancel, leaving $2\sin u\cos v$. Replacing $u$ and $v$ gives the first of four sum-to-product identities:

The last minus sign is easy to miss. It follows from subtracting the cosine expansions, whose sine-product terms have opposite signs. It can also be checked with $A=\pi$, $B=0$: the left side is −2, so the right side must be negative.

Use these identities when averaging the two arguments makes a familiar angle or when a product will reveal a common factor. The original angles do not need to be acute, and there are no denominator restrictions in these four sine/cosine identities. If you later divide by a factor, that new step must still be checked.

Read the two new angles separately. The average $(A+B)/2$ and half-difference $(A-B)/2$ are not the original sum and difference. Writing them before selecting the functions reduces arithmetic and sign errors.

Turn a product into a sum

The same compound-angle identities can be read in the opposite direction. Add $\sin(A+B)$ and $\sin(A-B)$: the $\cos A\sin B$ terms cancel, leaving $2\sin A\cos B$. Subtract instead, and the other mixed product remains. Applying the same idea to the cosine pair gives four product-to-sum relationships:

All four hold for real $A$ and $B$. There is no division by a variable expression in their derivation. If your starting product lacks the factor 2, multiply and divide by 2—or simply use half the corresponding right side. Keep that numerical factor visible until the transformation is complete.

The order in the last formula is especially useful to check: cosine of the difference comes first, then minus cosine of the sum. For example, putting $A=B=x$ gives $2\sin^2x=1-\cos2x$, exactly the double-angle rearrangement already established. This is a quick consistency check that also links the identity families.

Choose the direction that makes the next step simpler. A product may become a manageable sum of familiar angles; a sum may become a factor that cancels with a known nonzero denominator. There is no rule that every expression must end as a product or that every product should be expanded. The aim is an equivalent expression with a clear next step, while preserving every condition attached to the original calculation.

Choose an identity and keep the domain

Before expanding, look at the angle arguments. Are two angles being added? Is one twice another? Do two terms have a simple average? An identity becomes easier to choose when you see that structure before focusing on the coefficients. Then write the matching formula with its arguments clearly identified.

An identity is true for every input where its displayed expressions are defined. It is different from an equation that asks which particular inputs make an equality true. To prove an identity, transform one side into the other using valid equalities. You can also transform both sides into the same expression, provided you do not assume the desired equality as a starting fact.

Consider

Both sums have average 2 x and half-differencex. Sum-to-product gives

The original denominator must be nonzero. Since it equals $2\cos2x\cos x$, that condition ensures both factors are nonzero and justifies the cancellation. The simplified tangent may exist at some additional inputs, but those inputs do not become valid for the original fraction. For instance, at $x=\pi/2$ the original fraction is 0/0 even though $\tan2x=0$.

For exact-value questions, choose an angle split made from values you know. For proof questions, choose a form that resembles the target. After simplifying, check one ordinary allowed input to catch arithmetic or sign mistakes. Such a check is useful evidence about your working, but it does not replace an algebraic proof valid over the whole stated domain.