Let \(\alpha, \beta, \gamma\) be the three interior angles of a triangle. These identities involve sums and products of trigonometric functions of the triangle’s angles and are useful in solving geometric and trigonometric problems related to triangles.
\[ \alpha, \beta, \gamma \text{ satisfy } \alpha + \beta + \gamma = \pi \]
\[ \sin \alpha + \sin \beta + \sin \gamma = 4 \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2} \]
Explanation: The sum of sines of the triangle’s angles equals four times the product of cosines of half-angles.
\[ \cos \alpha + \cos \beta + \cos \gamma = 1 + 4 \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \sin \frac{\gamma}{2} \]
Explanation: The sum of cosines of the angles relates to the product of sines of half-angles plus one.
\[ \sin \alpha + \sin \beta - \sin \gamma = 4 \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \cos \frac{\gamma}{2} \]
Explanation: Combination of sines expressed via products of half-angle sines and cosines.
\[ \cos \alpha + \cos \beta - \cos \gamma = 4 \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \sin \frac{\gamma}{2} - 1 \]
Explanation: Sum and difference of cosines expressed using half-angle functions.
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \cos \alpha \cos \beta \cos \gamma + 2 \]
Explanation: Sum of squares of sines related to product of cosines.
\[ \sin^2 \alpha + \sin^2 \beta - \sin^2 \gamma = 2 \sin \alpha \sin \beta \cos \gamma \]
Explanation: Another relation involving squares of sines and product of sine and cosine functions.
\[ \sin 2\alpha + \sin 2\beta + \sin 2\gamma = 4 \sin \alpha \sin \beta \sin \gamma \]
Explanation: Sum of double-angle sines linked to product of sines of angles.
\[ \sin 2\alpha + \sin 2\beta - \sin 2\gamma = 4 \cos \alpha \cos \beta \sin \gamma \]
Explanation: Combination of double-angle sines expressed via products of cosines and sine.
\[ \tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma \]
Explanation: Tangent sum identity specific to triangle angles.
\[ \cot \tan \frac{\alpha}{2} + \cot \tan \frac{\beta}{2} + \cot \tan \frac{\gamma}{2} = \cot \tan \frac{\alpha}{2} \cot \tan \frac{\beta}{2} \cot \tan \frac{\gamma}{2} \]
Note: This involves nested cotangent and tangent functions of half angles and reflects complex angle relationships.
\[ \cot \tan \alpha \cot \tan \beta + \cot \tan \alpha \cot \tan \gamma + \cot \tan \beta \cot \tan \gamma = 1 \]
Note: Another advanced identity involving nested cotangent and tangent terms.