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If someone asks you for the exact value of cos⁡(75∘)\cos(75^\circ), your calculator gives an endless decimal like 0.258819…0.258819\dots, which loses mathematical precision. But notice that 75∘75^\circ is simply 45∘+30∘45^\circ + 30^\circ — two angles whose exact coordinates you already know.
It is tempting to guess that cos⁡(45∘+30∘)\cos(45^\circ + 30^\circ) equals cos⁡(45∘)+cos⁡(30∘)\cos(45^\circ) + \cos(30^\circ), but trigonometric functions do not distribute over addition. Around 150 CE, Greek-Egyptian astronomer Ptolemy needed precise chord lengths to map planetary movements, which led him to discover how sums of angles truly combine.
How do angle sums actually behave when we combine them inside sine and cosine?

The Sum and Difference Identities

The sum and difference identities allow you to rewrite trigonometric functions of combined angles like (A±B)(A \pm B) into products of single-angle functions. Cosine alternates function pairs and flips the sign, while sine mixes functions and preserves the sign.
Can we use these patterns to evaluate non-standard angles like 75∘75^\circ in exact radical form?

Finding Exact Values

To find cos⁡(75∘)\cos(75^\circ), break 75∘75^\circ into the sum of two standard special angles: 45∘45^\circ and 30∘30^\circ. Recall that on the unit circle, cos⁡(45∘)=22\cos(45^\circ) = \frac{\sqrt{2}}{2}, sin⁡(45∘)=22\sin(45^\circ) = \frac{\sqrt{2}}{2}, cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}, and sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}.
Substitute these values directly into the cosine sum identity: cos⁡(45∘+30∘)=cos⁡(45∘)cos⁡(30∘)−sin⁡(45∘)sin⁡(30∘)\cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)
A frequent error is writing cos⁡(45∘+30∘)=cos⁡(45∘)+cos⁡(30∘)=2+32≈1.57\cos(45^\circ + 30^\circ) = \cos(45^\circ) + \cos(30^\circ) = \frac{\sqrt{2} + \sqrt{3}}{2} \approx 1.57, which is impossible because cosine can never exceed 11.
What happens when an expression is already expanded and you need to simplify it backwards?