Trigonometric Identities — NDA Mathematics

Formulas (1)

• The fundamental identities · The three Pythagorean identities
  $$\sin^2\theta+\cos^2\theta=1,\quad \sec^2\theta-\tan^2\theta=1,\quad \csc^2\theta-\cot^2\theta=1$$

Reference tables (2)

Formulas (2)

• sin(A ± B) and cos(A ± B) · Sum and difference
  $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\quad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$
• tan(A ± B) and cot(A ± B) · Tangent of a sum/difference
  $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

Formulas (2)

• Double-angle formulas · The three forms of cos 2A
  $$\cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2 A-1$$
• Triple-angle formulas · Triple angle
  $$\sin 3A=3\sin A-4\sin^3 A,\qquad \cos 3A=4\cos^3 A-3\cos A$$

Formulas (3)

• Product-to-sum formulas · The four product-to-sum identities
  $$2\sin A\cos B=\sin(A+B)+\sin(A-B),\quad 2\cos A\cos B=\cos(A+B)+\cos(A-B)$$
• Sum-to-product formulas · Sum-to-product
  $$\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2},\quad \cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}$$
• Conditional identities (A + B + C = 90° or 180°) · The two signature conditional identities
  $$A+B+C=\pi:\ \tan A+\tan B+\tan C=\tan A\tan B\tan C$$

Formulas (1)

• AM-GM for reciprocal-type minima · AM-GM minimum
  $$u+v\ge 2\sqrt{uv}\ \ (u,v>0),\quad \text{equality at } u=v$$