Lesson 4-1: Trigonometric Identities – Derivation & Applications

Learning Objectives

Prove core trigonometric identities from first principles and geometric reasoning.
Transform expressions to equivalent forms using identity toolkits.
Apply sum/difference and double-angle formulas to simplify and evaluate.

Diagnose common pitfalls and design consistent proof strategies.
Model physical and signal phenomena using trigonometric identities.

Identity Toolkit

Pythagorean

\sin^2\theta+\cos^2\theta=1

Reciprocal

\sec\theta=\tfrac{1}{\cos\theta},\; \csc\theta=\tfrac{1}{\sin\theta},\; \cot\theta=\tfrac{1}{\tan\theta}

Quotient

\tan\theta=\tfrac{\sin\theta}{\cos\theta},\; \cot\theta=\tfrac{\cos\theta}{\sin\theta}

Co-Function

\sin(\tfrac{\pi}{2}-x)=\cos x,\; \tan(\tfrac{\pi}{2}-x)=\cot x

Sum/Diff (sin)

\sin(A\pm B)=\sin A\cos B \pm \cos A\sin B

Sum/Diff (cos)

\cos(A\pm B)=\cos A\cos B \- \sin A\sin B

Sum/Diff (tan)

\tan(A\pm B)=\tfrac{\tan A \pm \tan B}{1 \- \tan A\tan B}

Double-Angle

\sin 2x=2\sin x\cos x,\; \cos 2x=\cos^2 x-\sin^2 x=1-2\sin^2 x=2\cos^2 x-1

Key Derivations

Pythagorean Identity

On the unit circle, a point has coordinates $(\cos\theta,\sin\theta)$ so the radius condition yields

\cos^2\theta+\sin^2\theta=1

Sum/Difference (via rotation)

Rotations add: using matrix $R(\alpha)$ and $R(\beta)$ , we get

\sin(A+B)=\sin A\cos B+\cos A\sin B

\cos(A+B)=\cos A\cos B-\sin A\sin B

Double-Angle

Set $A=B=x$ in sum identities:

\sin 2x=2\sin x\cos x

\cos 2x=\cos^2 x-\sin^2 x=1-2\sin^2 x=2\cos^2 x-1

Product-to-Sum (outline)

From sum/difference, derive $\cos A\cos B=\tfrac{1}{2}[\cos(A+B)+\cos(A-B)]$ etc.

\cos A\cos B=\tfrac{1}{2}[\cos(A+B)+\cos(A-B)]

\sin A\sin B=\tfrac{1}{2}[\cos(A-B)-\cos(A+B)]

Worked Examples

Example 1: Simplify using identities

$E=\tfrac{\sin 2x}{1+\cos 2x}\cdot \cot x$ .

Show Solution

Use $\sin 2x=2\sin x\cos x$ , $1+\cos 2x=2\cos^2 x$ :

E=\tfrac{2\sin x\cos x}{2\cos^2 x}\cdot \tfrac{\cos x}{\sin x}=1

Example 2: Prove identity

Prove $\sin^4 x+\cos^4 x=1-2\sin^2 x\cos^2 x$ .

Show Proof

$(\sin^2 x+\cos^2 x)^2=\sin^4 x+2\sin^2 x\cos^2 x+\cos^4 x=1$

Rearrange: $\sin^4 x+\cos^4 x=1-2\sin^2 x\cos^2 x$ .

Example 3: Evaluate expression

Compute $\cos 75^\circ$ using sum/difference.

Show Solution

$75^\circ=45^\circ+30^\circ$ :

\cos 75^\circ=\cos 45^\circ\cos 30^\circ-\sin 45^\circ\sin 30^\circ=\tfrac{\sqrt{2}}{2}\cdot\tfrac{\sqrt{3}}{2}-\tfrac{\sqrt{2}}{2}\cdot\tfrac{1}{2}=\tfrac{\sqrt{6}-\sqrt{2}}{4}

Example 4: Solve for angle

Given $\sin 2x=\tfrac{\sqrt{3}}{2}$ , find all solutions in $[0,2\pi)$ .

Show Solution

$2x=\tfrac{\pi}{3}+2k\pi,\; 2x=\tfrac{2\pi}{3}+2k\pi$ ⇒ $x=\tfrac{\pi}{6}+k\pi,\; x=\tfrac{\pi}{3}+k\pi$ .

Strategy & Common Pitfalls

Strategy

Convert everything to $\sin x, \cos x$ when stuck.
Use conjugates to handle expressions like $1\pm \sin x$ or $1\pm \cos x$ .
Prefer identities that reduce powers: e.g., use $\cos 2x$ to eliminate $\sin^2 x$ or $\cos^2 x$ .
Track domains to avoid dividing by zero.

Pitfalls

Misusing $\sin(A+B)$ as $\sin A+\sin B$ (incorrect).
Dropping absolute values when inverting functions or taking square roots.
Ignoring angle units (degrees vs. radians).
Forgetting extraneous solutions when multiplying by factors like $\cos x$ .

Practice Problems

Problem 1

Prove sin^2 x = (1 - cos 2x)/2 and cos^2 x = (1 + cos 2x)/2.

Show Solution

Use cos 2x = cos^2 x - sin^2 x and cos^2 x + sin^2 x = 1; solve the 2×2 system.

Problem 2

Simplify (1 - cos x)/(sin x) - (sin x)/(1 + cos x).

Show Solution

Use conjugate on the first term or common denominator: result is 0.

Problem 3

Show tan(A) + tan(B) = (sin(A + B))/(cos A cos B) when A + B ≠ (π/2) + kπ.

Show Solution

Use tan = sin/cos and the sum formula for sin(A+B).

Problem 4

Prove 1 + tan^2 x = sec^2 x.

Show Solution

Divide sin^2 + cos^2 = 1 by cos^2.

Problem 5

Given sin α = 3/5, cos β = 4/5 with α, β in first quadrant, compute cos(α + β).

Show Solution

Find cos α = 4/5, sin β = 3/5, then apply cos sum formula.

Problem 6

Derive sin 3x and cos 3x using angle addition repeatedly.

Show Solution

Use sin(2x + x), cos(2x + x); express via sin x, cos x.

Extended Identities Library

Identity Case 1: Conjugate Simplification

Multiply by the conjugate to remove 1−cos x in numerator.

\frac{1-\cos x}{\sin x}

=\frac{(1-\cos x)(1+\cos x)}{\sin x(1+\cos x)}

=\frac{\sin^2 x}{\sin x(1+\cos x)}

=\frac{\sin x}{1+\cos x}

Identity Case 2: Quotient to Tangent

Use conjugate on denominator and Pythagorean identity.

\frac{\sin x}{\cos x+1}

=\frac{\sin x(\cos x-1)}{(\cos x+1)(\cos x-1)}

=\frac{-\sin x(1-\cos x)}{1-\cos^2 x}

=-\frac{1-\cos x}{\sin x}

Identity Case 3: Product-to-Sum

Apply product-to-sum identity.

2\sin A\cos B

=\sin(A+B)+\sin(A-B)

Identity Case 4: Sum-to-Product

Use sum-to-product to factor cosine difference.

\cos A-\cos B

=-2\sin\left(\tfrac{A+B}{2}\right)\sin\left(\tfrac{A-B}{2}\right)

Identity Case 5: Half-Angle (Sine)

Half-angle reduces powers.

\sin^2 x

=\tfrac{1-\cos 2x}{2}

Identity Case 6: Half-Angle (Cosine)

Use power-reduction for integrals/summations.

\cos^2 x

=\tfrac{1+\cos 2x}{2}

Identity Case 7: Double-Angle to Tangent

Divide by cos^2 x and use tan = sin/cos.

\cos 2x

=\frac{1-\tan^2 x}{1+\tan^2 x}

Identity Case 8: Reciprocal Pair

Divide sin^2+cos^2=1 by cos^2.

1+\tan^2 x

=\sec^2 x

Identity Case 9: Canonical Form via Sum

Apply sum-to-product to numerator and denominator.

\frac{\sin x+\sin y}{\cos x+\cos y}

=\frac{2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2}}{2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2}}

=\tan\tfrac{x+y}{2}

Identity Case 10: Rationalizing with t-substitution

Universal t-substitution for rational trig forms.

\tan\frac{x}{2}=t

\Rightarrow \sin x=\frac{2t}{1+t^2},\; \cos x=\frac{1-t^2}{1+t^2}

Identity Case 11: Angle Addition (Cosine)

From rotation matrices or unit-circle geometry.

\cos(\alpha+\beta)

=\cos\alpha\cos\beta-\sin\alpha\sin\beta

Identity Case 12: Auxiliary-Angle Method

Rewrite linear combo as phase-shifted sinusoid.

a\sin x+b\cos x

=R\sin(x+\phi),\; R=\sqrt{a^2+b^2}

Applications & Modeling

Signal Phasors

Sum of sinusoids with same frequency can be expressed as a single sinusoid using sum identities.

a\sin(\omega t)+b\cos(\omega t)=R\sin(\omega t+\phi),\; R=\sqrt{a^2+b^2}

Useful in AC circuits and oscillations.

Harmonic Identities

Double-angle helps reduce integrands or derive envelope relationships in physics.

\sin^2 x=\tfrac{1-\cos 2x}{2},\; \cos^2 x=\tfrac{1+\cos 2x}{2}

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