Trigonometry

Trigonometric identities

Trigonometric identities are equations true for all admissible values of the variables involving trigonometric ratios, serving as the cornerstone for algebraic simplification and calculus integration.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Pythagorean Trigonometric Foundation

For all \theta \in \mathbb{R} (with appropriate domains for denominators and tangents), \sin^2 \theta + \cos^2 \theta = 1, \quad 1 + \tan^2 \theta = \sec^2 \theta, \quad 1 + \cot^2 \theta = \csc^2 \theta.

📐 Geometric Interpretation

Derived directly from the unit circle equation $x^2 + y^2 = 1$ where $x = \cos \theta$ and $y = \sin \theta$, representing the Euclidean distance from the origin to any point on the circumference.

⚙️ Algebraic & Structural Meaning

Provides the primary reduction mechanism for converting high-degree trigonometric polynomials into linear or lower-degree rational forms, essential for calculus transformations.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Domain Restriction for Tangent and Secant	`\theta \neq \frac{\pi}{2} + k\pi, \; k \in \mathbb{Z}`	Strictly Mandatory	Division by zero occurs, rendering the identity undefined and resulting in vertical asymptotes.
Domain Restriction for Cotangent and Cosecant	`\theta \neq k\pi, \; k \in \mathbb{Z}`	Strictly Mandatory	Leads to undefined expressions due to division by \sin \theta = 0.
Argument Continuity and Periodicity	`\theta \in \mathbb{R}`	Essential for functional transformations	Ignoring periodicity leads to loss of infinite sets of general solutions in trigonometric equations.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Sum of Sines/Cosines in Arithmetic Progression

$$ \sum_{k=1}^{n} \sin(a + k\cdot d) = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \sin\left(a + \frac{(n+1)d}{2}\right) $$

Rule: Use the multiplicative angle formula to sum series in seconds without expanding term by term.

Application: Frequently asked in UP PGT for summing sine and cosine series with common angle differences.

Maximum and Minimum of $a \cos \theta + b \sin \theta$

$$ -\sqrt{a^2 + b^2} \le a \cos \theta + b \sin \theta \le \sqrt{a^2 + b^2} $$

Rule: Instantly bound expressions of linear combinations of sine and cosine by taking the square root of the sum of squares.

Conditional Trigonometric Identity for Triangles

$$ \text{If } A+B+C = \pi, \text{ then } \tan A + \tan B + \tan C = \tan A \tan B \tan C $$

Rule: Apply immediately when interior angles of a triangle are given, bypassing long algebraic expansions.

Triple Angle Product Identities

$$ \sin \theta \sin(60^\circ - \theta) \sin(60^\circ + \theta) = \frac{1}{4} \sin(3\theta) $$

Rule: Recognize product patterns to instantly substitute into single-term multiple-angle expressions.

Application: Used to evaluate numerical trigonometric products rapidly in objective papers.

Half-Angle Substitution (Weierstrass Substitution)

$$ \sin \theta = \frac{2t}{1+t^2}, \quad \cos \theta = \frac{1-t^2}{1+t^2}, \quad \text{where } t = \tan\left(\frac{\theta}{2}\right) $$

Rule: Convert rational trigonometric functions into algebraic rational functions of $t$.

Value Substitution (Testing $\theta = 0^\circ$ or $45^\circ$)

$$ \text{Substitute } \theta = \frac{\pi}{4} \text{ to eliminate variables in identity verification.} $$

Rule: If an identity holds universally, it holds for specific standard angles. Check options by substituting $\theta = 0^\circ$ or $45^\circ$.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Extraneous Solutions in Squaring Identities

Common Pitfall: Squaring both sides of a trigonometric equation to eliminate radicals introduces extraneous roots that do not satisfy the original equation.

Counterexample: \sin \theta + \cos \theta = \sqrt{2} \implies (\sin \theta + \cos \theta)^2 = 2 \implies 1 + \sin(2\theta) = 2

Why it fails: Squaring changes signs and expands the solution set; solutions where $\sin \theta + \cos \theta = -\sqrt{2}$ are incorrectly included.

🛡️ Exam Defense: Always cross-check final angle solutions against the original unsquared equation's domain and sign constraints.

⚠️ COMMON ERROR #02

Ignoring Quadrant Signs in Square Roots

Common Pitfall: Writing $\sqrt{\cos^2 \theta} = \cos \theta$ universally without checking the quadrant.

Counterexample: \sqrt{\cos^2 \theta} = |\cos \theta|

Why it fails: In quadrants II and III, $\cos \theta$ is negative, so dropping the absolute value leads to sign errors.

🛡️ Exam Defense: Always use absolute value bars when simplifying square roots of squared trigonometric functions: $\sqrt{1-\sin^2 \theta} = |\cos \theta|$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Pythagorean Identities	\theta \in \mathbb{R}	\sin^2\theta + \cos^2\theta = 1	Radius of unit circle is 1	Sum of squared sin/cos terms
Compound Angle Formulas	Universal domain	\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B	Rotation of coordinate axes	Sums of angles inside trig functions
Multiple Angle Formulas	Universal domain	\cos(2\theta) = 2\cos^2\theta - 1	Scaling frequency of periodic wave	Double or triple angle terms
Transformation Formulas	Universal domain	2\sin A \cos B = \sin(A+B) + \sin(A-B)	Converting products to sums	Product of two different trig functions

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $\tan \theta + \sec \theta = e^x$, find the value of $\cos \theta$.

(A) \frac{e^x + e^{-x}}{2}(B) \frac{2e^x}{e^{2x} + 1}(C) \frac{e^x - e^{-x}}{e^x + e^{-x}}(D) \frac{1}{e^x + e^{-x}}

📚 Method A: Formal Textbook Derivation

Step 1: Use the identity $\sec^2 \theta - \tan^2 \theta = 1 \implies (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1$. Step 2: Since $\sec \theta + \tan \theta = e^x$, we have $\sec \theta - \tan \theta = e^{-x}$. Step 3: Add both equations to get $2 \sec \theta = e^x + e^{-x} \implies \sec \theta = \frac{e^x + e^{-x}}{2}$. Step 4: Hence, $\cos \theta = \frac{2}{e^x + e^{-x}} = \frac{2e^x}{e^{2x} + 1}$.

⚡ Method B: 25-Second Exam Shortcut

Use reciprocal property: if $x + \frac{1}{x} = K$, then extraction of cosine via secant reciprocal yields option (B) instantly.

💡 Tip: Whenever $\sec \theta \pm \tan \theta$ or $\csc \theta \pm \cot \theta$ is given as an exponential, immediately write its reciprocal for the conjugate pair.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the maximum value of $3 \sin \theta + 4 \cos \theta + 5$?

(A) 5(B) 10(C) 12(D) \sqrt{50}

📚 Method A: Formal Textbook Derivation

Step 1: Identify $a = 3$ and $b = 4$ in $a\cos\theta + b\sin\theta$. Step 2: Calculate maximum value of the trigonometric part as $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. Step 3: Add the constant 5 to get $5 + 5 = 10$.

⚡ Method B: 25-Second Exam Shortcut

Max value = $\sqrt{3^2 + 4^2} + 5 = 5 + 5 = 10$. Done in 5 seconds.

💡 Tip: Do not forget to add or subtract any constant term appended outside the linear sine-cosine combination.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The value of $\sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ$ is:

A 8

B 9

C 9.5

D 10

Derivation & Analysis:

Pair terms using complementary angles $\sin^2(90^\circ - \theta) = \cos^2 \theta$. There are 18 terms from $5^\circ$ to $90^\circ$. Pairing $5^\circ$ with $85^\circ$ gives 1, plus $\sin^2 90^\circ = 1$ and $\sin^2 45^\circ = 1/2$, totaling $8 + 1 + 0.5 = 9.5$.

MCQ 2: If $\cos \alpha + \cos \beta = 0 = \sin \alpha + \sin \beta$, then $\cos(2\alpha - 2\beta)$ is equal to:

A 0

B 1

C -1

D $\frac{1}{2}$

Derivation & Analysis:

From $\cos \alpha = -\cos \beta$ and $\sin \alpha = -\sin \beta$, squaring and adding gives no direct contradiction, but dividing gives $\tan \alpha = \tan \beta \implies \alpha - \beta = \pi/2 + k\pi$. Thus $2\alpha - 2\beta = \pi + 2k\pi$, so $\cos(2\alpha - 2\beta) = -1$ or check consistency. Wait, $\cos \alpha + \cos \beta = 0 \implies \alpha - \beta = \pi$ or odd multiple of $\pi$. Thus $2\alpha - 2\beta$ is an even multiple of $\pi$, making $\cos(0) = 1$.

MCQ 3: The expression $\frac{\sin 3\theta}{\sin \theta} - \frac{\cos 3\theta}{\cos \theta}$ is identically equal to:

A 1

B 2

C 4

D $\sin \theta \cos \theta$

Derivation & Analysis:

Combine fractions over common denominator $\sin \theta \cos \theta$: $\frac{\sin 3\theta \cos \theta - \cos 3\theta \sin \theta}{\sin \theta \cos \theta} = \frac{\sin(3\theta - \theta)}{\sin \theta \cos \theta} = \frac{\sin 2\theta}{\frac{1}{2}\sin 2\theta} = 2$.

MCQ 4: If $\tan \theta = \frac{1}{2}$ and $\tan \phi = \frac{1}{3}$, then the value of $\theta + \phi$ is:

A $\frac{\pi}{6}$

B $\frac{\pi}{4}$

C $\frac{\pi}{3}$

D $\frac{\pi}{2}$

Derivation & Analysis:

Use $\tan(\theta + \phi) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2}\cdot\frac{1}{3}} = \frac{5/6}{5/6} = 1$. Since $\theta, \phi \in (0, \pi/2)$, $\theta + \phi = \frac{\pi}{4}$.

MCQ 5: The maximum value of $\sin^6 \theta + \cos^6 \theta$ is:

A 1

B $\frac{3}{4}$

C $\frac{1}{2}$

D $\frac{1}{4}$

Derivation & Analysis:

Using identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$, we get $(\sin^2 \theta + \cos^2 \theta)((\sin^2 \theta + \cos^2 \theta)^2 - 3\sin^2 \theta \cos^2 \theta) = 1 - \frac{3}{4}\sin^2(2\theta)$. Maximum occurs when $\sin(2\theta) = 0$, giving $1 - 0 = 1$.

MCQ 6: If $x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta$ and $x \sin \theta - y \cos \theta = 0$, then $x^2 + y^2$ is equal to:

A 1

B 2

C $\frac{1}{2}$

D 4

Derivation & Analysis:

From $x \sin \theta = y \cos \theta$, substitute into the first equation: $(y \cos \theta)\sin^2 \theta + (x \sin \theta)\cos^2 \theta = \sin \theta \cos \theta \implies x^2 + y^2 = 1$ after algebraic reduction.

MCQ 7: General solution of $\sqrt{3} \cos \theta + \sin \theta = \sqrt{2}$ is:

A $n\pi + (-1)^n \frac{\pi}{4} - \frac{\pi}{3}$

B $2n\pi \pm \frac{\pi}{4}$

C $n\pi + (-1)^n \frac{\pi}{3} + \frac{\pi}{4}$

D $2n\pi \pm \frac{\pi}{3} - \frac{\pi}{4}$

Derivation & Analysis:

Divide by 2: $\frac{\sqrt{3}}{2}\cos \theta + \frac{1}{2}\sin \theta = \frac{1}{\sqrt{2}} \implies \cos\left(\theta - \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{4}\right)$, yielding $\theta - \frac{\pi}{3} = 2n\pi \pm \frac{\pi}{4}$, or general form equivalent.

MCQ 8: Assertion (A): $\sec^2 \theta + \csc^2 \theta$ can never be less than 4. Reason (R): Arithmetic Mean is greater than or equal to Geometric Mean.

A Both (A) and (R) are true and (R) is the correct explanation of (A)

B Both (A) and (R) are true but (R) is not the correct explanation of (A)

C (A) is true but (R) is false

D (A) is false but (R) is true

Derivation & Analysis:

By AM-GM inequality, $\tan^2 \theta + \cot^2 \theta \ge 2$. Adding 2 to both sides gives $\tan^2 \theta + 2 + \cot^2 \theta \ge 4 \implies \sec^2 \theta + \csc^2 \theta \ge 4$. Thus both assertion and reason are correct and linked.

MCQ 9: If $\cos A = \frac{3}{5}$ and $\cos B = \frac{5}{13}$, where $A$ and $B$ are acute angles, then $\cos(A - B)$ is:

A $\frac{33}{65}$

B $\frac{63}{65}$

C $\frac{16}{65}$

D $\frac{56}{65}$

Derivation & Analysis:

Since $\cos A = 3/5 \implies \sin A = 4/5$, and $\cos B = 5/13 \implies \sin B = 12/13$. $\cos(A-B) = \cos A \cos B + \sin A \sin B = \frac{3}{5}\cdot\frac{5}{13} + \frac{4}{5}\cdot\frac{12}{13} = \frac{15 + 48}{65} = \frac{63}{65}$.

Q10

MCQ 10: The value of $\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ$ is:

A 1

B 2

C 4

D 8

Derivation & Analysis:

Rewrite using cotangent conversions: $\tan 9^\circ + \cot 9^\circ - (\tan 27^\circ + \cot 27^\circ) = \frac{2}{\sin 18^\circ} - \frac{2}{\sin 54^\circ} = \frac{2}{\sin 18^\circ} - \frac{2}{\cos 36^\circ} = 4$.

Common questions

Squaring both sides or multiplying by expressions containing variables expands the domain of the equation, introducing roots that satisfy the squared equation but violate the sign constraints of the original unsquared relation.

Failure to check domain restrictions (such as denominators becoming zero for tan, cot, sec, csc) and missing general solution periodic bounds.

Mastering standard shortcuts like the $a\cos\theta + b\sin\theta$ bounds, substituting standard angle values (like $45^\circ$) to test options, and memorizing standard product-to-sum expansions.

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