Formulas

Trig Identities

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Sum-to-Product Identities

  • Angles (arguments of functions): \(\alpha\), \(\beta\)
    Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha\)
    1. Sum of sines
      \(\sin \alpha + \sin \beta =\) \( 2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize\)
    2. Difference of sines
      \(\sin \alpha – \sin \beta =\) \( 2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize\)
    3. Sum of cosines
      \(\cos \alpha + \cos \beta =\) \( 2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize\)
    4. Difference of cosines
      \(\cos \alpha – \cos \beta =\) \( -2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize\)
    5. Sum of tangents
      \(\tan\alpha + \tan \beta =\) \( \large\frac{{\sin \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize\)
    6. Difference of tangents
      \(\tan\alpha – \tan \beta =\) \( \large\frac{{\sin \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize\)
    7. Sum of cotangents
      \(\cot\alpha + \cot \beta =\) \( \large\frac{{\sin \left( {\beta + \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize\)
    8. Difference of cotangents
      \(\cot\alpha – \cot \beta =\) \( \large\frac{{\sin \left( {\beta – \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize\)
    9. Sum of cosine and sine
      \(\cos\alpha + \sin \alpha =\) \( \sqrt 2 \cos \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =\) \( \sqrt 2 \sin\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)\)
    10. Difference of cosine and sine
      \(\cos\alpha – \sin \alpha =\) \( \sqrt 2 \sin \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =\) \( \sqrt 2 \cos\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)\)
    11. Sum of tangent and cotangent
      \(\tan\alpha + \cot \beta =\) \( \large\frac{{\cos \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize\)
    12. Difference of tangent and cotangent
      \(\tan\alpha – \cot \beta =\) \( -\large\frac{{\cos \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize\)
    13. \(1 + \cos \alpha = 2\,{\cos ^2}\large\frac{\alpha }{2}\normalsize\)
    14. \(1 – \cos \alpha = 2\,{\sin ^2}\large\frac{\alpha }{2}\normalsize\)
    15. \(1 + \sin \alpha =\) \( 2\,{\cos ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)\)
    16. \(1 – \sin \alpha =\) \( 2\,{\sin ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)\)