These formulas convert sums or differences of sine and cosine functions into products of trigonometric functions, simplifying the expressions and aiding in solving equations.
\[ \cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} \]
Explanation: The sum of two cosines is expressed as twice the product of cosines of half-sum and half-difference of the angles.
\[ \cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} \]
Explanation: The difference of two cosines is represented as negative twice the product of sines of half-sum and half-difference.
\[ \sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} \]
Explanation: The sum of two sines becomes twice the product of sine of half-sum and cosine of half-difference.
\[ \sin \alpha - \sin \beta = 2 \cos \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} \]
Explanation: The difference of two sines equals twice the product of cosine of half-sum and sine of half-difference.
\[ \sin \alpha + \cos \alpha = \sqrt{2} \sin \left(\alpha + \frac{\pi}{4}\right) = \sqrt{2} \cos \left(\frac{\pi}{4} - \alpha\right) \]
Explanation: Sum of sine and cosine of the same angle is rewritten using sine or cosine shifted by \(\pi/4\) (45°) with amplitude \(\sqrt{2}\).
\[ \sin \alpha - \cos \alpha = \sqrt{2} \sin \left(\alpha - \frac{\pi}{4}\right) = -\sqrt{2} \cos \left(\frac{\pi}{4} - \alpha\right) \]
Explanation: Difference of sine and cosine of the same angle can also be expressed similarly with phase shift and amplitude.
\[ \tan \alpha + \tan \beta = \frac{\sin(\alpha + \beta)}{\cos \alpha \cos \beta} \]
Explanation: Sum of tangents is expressed as sine of sum divided by the product of cosines.
\[ \tan \alpha - \tan \beta = \frac{\sin(\alpha - \beta)}{\cos \alpha \cos \beta} \]
Explanation: Difference of tangents equals sine of difference divided by product of cosines.
\[ \cot \alpha + \cot \beta = \frac{\sin(\alpha - \beta)}{\sin \alpha \sin \beta} \]
Explanation: Sum of cotangents is sine of difference over product of sines.
\[ \cot \alpha - \cot \beta = \frac{\sin(\beta - \alpha)}{\sin \alpha \sin \beta} \]
Explanation: Difference of cotangents is sine of difference (inverted) over product of sines.
\[ \tan \alpha + \cot \alpha = 2 \cos \sec 2\alpha \]
Explanation: The sum of tangent and cotangent of the same angle relates to secant and cosine of double angle.
\[ \tan \alpha - \cot \alpha = -2 \cot \tan 2\alpha \]
Explanation: Difference of tangent and cotangent connects to cotangent and tangent of double angle.