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)

Watch out for (4)

• It's $1+\tan^2\theta=\sec^2\theta$, never $1-\tan^2\theta$
  → The fundamental identities
• Don't swap sin and cos at 30° and 60°
  → Standard-angle values and allied reductions
• An allied angle can FLIP the sign — don't assume the ratio stays positive
  → Signs by quadrant (ASTC) and reductions
• One ratio fixes magnitudes only — the QUADRANT fixes the sign
  → All ratios from one ratio and a quadrant

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
  $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

Watch out for (3)

• $\sin(A+B)\neq\sin A+\sin B$ — sine is NOT linear
  → sin(A ± B) and cos(A ± B)
• The $\tan(A+B)$ denominator is $1-\tan A\tan B$, and the signs OPPOSE
  → tan(A ± B) and cot(A ± B)
• $\cos(A+B)\cos(A-B)=\cos^2 A-\sin^2 B$, not $\cos^2 A-\cos^2 B$
  → Conditional ratio manipulation (a sin²+b cos²=c, etc.)

Formulas (3)

• Double-angle formulas · Double-angle formulas
  $$\sin 2A=2\sin A\cos A,\qquad \cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2 A-1,\qquad \tan 2A=\dfrac{2\tan A}{1-\tan^2 A}$$
• Triple-angle formulas · Triple-angle formulas
  $$\sin 3A=3\sin A-4\sin^3 A,\qquad \cos 3A=4\cos^3 A-3\cos A,\qquad \tan 3A=\dfrac{3\tan A-\tan^3 A}{1-3\tan^2 A}$$
• Half-angle formulas and 1 ± cos A / 1 ± sin A · Half-angle formulas
  $$\sin\tfrac A2=\pm\sqrt{\tfrac{1-\cos A}{2}},\qquad \cos\tfrac A2=\pm\sqrt{\tfrac{1+\cos A}{2}},\qquad \tan\tfrac A2=\dfrac{1-\cos A}{\sin A}=\dfrac{\sin A}{1+\cos A}$$

Watch out for (3)

• $\sin 2A=2\sin A\cos A$ — not $2\sin A$, and $(\sin A)^2\neq\sin 2A$
  → Double-angle formulas
• Watch the sign pattern: $\sin 3A=3\sin A-4\sin^3 A$ but $\cos 3A=4\cos^3 A-3\cos A$
  → Triple-angle formulas
• The $\pm$ on a half-angle is fixed BY the quadrant of $A/2$, not free
  → Half-angle formulas and 1 ± cos A / 1 ± sin A

Formulas (3)

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

Watch out for (3)

• $2\sin A\sin B=\cos(A-B)-\cos(A+B)$ — cosines, and the difference comes FIRST
  → Product-to-sum formulas
• Half-SUM and half-DIFFERENCE go in fixed slots — don't swap them
  → Sum-to-product formulas
• $\tan A+\tan B+\tan C=\tan A\tan B\tan C$ only when $A+B+C=180°$
  → Conditional identities (A + B + C = 90° or 180°)

Formulas (1)

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

Watch out for (1)

• Max of $a\sin x+b\cos x$ is $\sqrt{a^2+b^2}$, NOT $a+b$
  → Range of a·sin x + b·cos x