Trigonometric Identities — JEE Mains Mathematics

Formulas (3)

• Signs by quadrant · Pythagorean identities
  $$\sin^2x+\cos^2x=1,\quad1+\tan^2x=\sec^2x,\quad1+\cot^2x=\csc^2x$$
• Sine and cosine of a sum · Compound angles
  $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\quad\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$
• Tangent of a sum · Tangent of a sum
  $$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

Watch out for (3)

• The triangle gives the size, not the sign
  → Signs by quadrant
• The sign in cos(A ± B) flips
  → Sine and cosine of a sum
• One tangent, two angles
  → Tangent of a sum

Formulas (2)

• Exact values at 15°, 18° and 36° · Values at 18° and 36°
  $$\sin18^\circ=\frac{\sqrt5-1}{4},\qquad\cos36^\circ=\frac{\sqrt5+1}{4}$$
• Double and triple angles · Double and triple angles
  $$\cos2\theta=1-2\sin^2\theta,\quad\sin3\theta=3\sin\theta-4\sin^3\theta,\quad\cos3\theta=4\cos^3\theta-3\cos\theta$$

Watch out for (2)

• sin 18° and cos 36° differ in one sign
  → Exact values at 15°, 18° and 36°
• sin 3θ and cos 3θ have opposite sign patterns
  → Double and triple angles

Formulas (2)

• Reducing through sin²θ cos²θ · Power sums
  $$\sin^4\theta+\cos^4\theta=1-2p,\quad\sin^6\theta+\cos^6\theta=1-3p,\quad p=\sin^2\theta\cos^2\theta$$
• A condition that fixes sin²θ · The perfect-square condition
  $$a\sin^4\theta+b\cos^4\theta=\frac{ab}{a+b}\ \Rightarrow\ \sin^2\theta=\frac{b}{a+b}$$

Watch out for (2)

• p runs from 0 to 1/4
  → Reducing through sin²θ cos²θ
• sin²θ takes the other coefficient
  → A condition that fixes sin²θ

Formulas (3)

• Cosine products with doubling angles · Doubling product
  $$\cos\theta\cos2\theta\cos4\theta\cdots\cos2^{n-1}\theta=\frac{\sin2^n\theta}{2^n\sin\theta}$$
• The 60° product identities · The 60° identity
  $$\sin\theta\sin(60^\circ-\theta)\sin(60^\circ+\theta)=\tfrac14\sin3\theta$$
• Sum-to-product and telescoping · Sum to product
  $$\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2},\quad\cos C+\cos D=2\cos\frac{C+D}{2}\cos\frac{C-D}{2}$$

Watch out for (3)

• Check where the last angle lands
  → Cosine products with doubling angles
• The angles must fit the pattern
  → The 60° product identities
• Multiply by the right sine
  → Sum-to-product and telescoping

Formulas (2)

• The bound on a sin θ + b cos θ · The bound
  $$-\sqrt{a^2+b^2}\le a\sin\theta+b\cos\theta\le\sqrt{a^2+b^2}$$
• Ranges through a bounded quantity · The range of p
  $$\sin^2\theta\cos^2\theta=\tfrac14\sin^22\theta\in\left[0,\tfrac14\right]$$

Watch out for (2)

• The bound needs one angle
  → The bound on a sin θ + b cos θ
• Check that each end is reached
  → Ranges through a bounded quantity