Properties of Inverse Trigonometric Functions - (Maths) for JEE Main

There are a few properties of inverse trigonometric functions which are crucial for problem solving and for a deeper conceptual understanding. Inverse trigonometric functions (also called arc-functions) give the angle (or arc length) whose trigonometric ratio has a given value. Each inverse trigonometric function is defined with a specific principal branch (range) so that it is single-valued. The domain of an inverse trigonometric function is the set of input values for which the inverse is defined; the range (or principal value) is the set of output angles returned by the inverse function.

Basic definitions, domains and principal values

Principal branches (ranges) used throughout

• arcsin x or sin^-1x: domain x ∈ [-1,1]; range y ∈ [-π/2, π/2].
• arccos x or cos^-1x: domain x ∈ [-1,1]; range y ∈ [0, π].
• arctan x or tan^-1x: domain x ∈ (-∞, ∞); range y ∈ (-π/2, π/2).
• arccot x or cot^-1x: domain x ∈ (-∞, ∞); range y ∈ (0, π).
• arcsec x or sec^-1x: domain |x| ≥ 1; range y ∈ [0, π], y ≠ π/2.
• arccosec x or cosec^-1x: domain |x| ≥ 1; range y ∈ [-π/2, π/2], y ≠ 0.

Fundamental composition identities

• sin(sin^-1x) = x for -1 ≤ x ≤ 1.
• cos(cos^-1x) = x for -1 ≤ x ≤ 1.
• tan(tan^-1x) = x for all real x.
• cot(cot^-1x) = x for all real x.
• sec(sec^-1x) = x for |x| ≥ 1.
• cosec(cosec^-1x) = x for |x| ≥ 1.
• sin^-1(sin y) = y for y ∈ [-π/2, π/2].
• cos^-1(cos y) = y for y ∈ [0, π].
• tan^-1(tan y) = y for y ∈ (-π/2, π/2).
• cot^-1(cot y) = y for y ∈ (0, π).
• sec^-1(sec y) = y for y ∈ [0, π], y ≠ π/2.
• cosec^-1(cosec y) = y for y ∈ [-π/2, π/2], y ≠ 0.

Important properties of inverse trigonometric functions

Property Set 1 - Reciprocal relations

• sin^-1x = cosec^-1(1/x), for x ∈ [-1,1] \ {0} (the right-hand side is defined because |1/x| ≥ 1).
• cos^-1x = sec^-1(1/x), for x ∈ [-1,1] \ {0}.
• tan^-1x = cot^-1(1/x) when x > 0; when x < 0 a shift by ±π may be required to match principal values (use principal ranges to decide the exact 
• cot^-1x = tan^-1(1/x) when x > 0; when x < 0 a shift by ±π may be required to match principal 

Property Set 2 - Odd/even and sign-change relations

• sin^-1(-x) = - sin^-1(x) for x ∈ [-1,1].
• tan^-1(-x) = - tan^-1(x) for all real x.
• cos^-1(-x) = π - cos^-1(x) for x ∈ [-1,1].
• cosec^-1(-x) = - cosec^-1(x) for |x| ≥ 1.
• sec^-1(-x) = π - sec^-1(x) for |x| ≥ 1.
• cot^-1(-x) = π - cot^-1(x) for real x.

Proofs for typical sign-change identities

Proof - sin^-1(-x) = - sin^-1(x)

Let sin^-1(-x) = y, so -x = sin y.

Then x = - sin y = sin(-y).

Therefore sin^-1(x) = -y = - sin^-1(-x).

Hence sin^-1(-x) = - sin^-1(x).

• Similarly, tan^-1(-x) = - tan^-1(x) and cosec^-1(-x) = - cosec^-1(x) (within respective domains) follow by the oddness of sine, tangent and cosecant on their principal intervals.

Proof - cos^-1(-x) = π - cos^-1(x)

Let cos^-1(-x) = y, so -x = cos y.

Then x = -cos y = cos(π - y).

Hence cos^-1(x) = π - y = π - cos^-1(-x).

Rearranging gives cos^-1(-x) = π - cos^-1(x).

• Similarly, sec^-1(-x) = π - sec^-1(x) and cot^-1(-x) = π - cot^-1(x) follow by corresponding symmetry in principal ranges.

Property Set 3 - Reciprocal input relations

• sin^-1(1/x) = cosec^-1x for |x| ≥ 1.
• cos^-1(1/x) = sec^-1x for |x| ≥ 1.
• Relations between tan^-1(1/x) and cot^-1(x) depend on the sign of x and principal branches; use principal-value conventions when converting.

Proof - sin^-1(1/x) = cosec^-1x for |x| ≥ 1

Let cosec^-1x = y, so x = cosec y.

Then 1/x = sin y.

Hence sin^-1(1/x) = y = cosec^-1x.

Illustrative examples (reciprocal relations)

• sin^-1(1/3) = cosec^-1(3).
• cos^-1(1/4) = sec^-1(4).
• tan^-1(-3) relates to cot^-1(-1/3) with a π-shift depending on principal values; check ranges before finalising such conversions.

Property Set 4 - Mixed-function arguments

• sin^-1(cos θ) = π/2 - θ if θ ∈ [0, π].
• cos^-1(sin θ) = π/2 - θ if θ ∈ [-π/2, π/2].
• tan^-1(cot θ) = π/2 - θ if θ ∈ (0, π).
• cot^-1(tan θ) = π/2 - θ if θ ∈ (-π/2, π/2).
• sec^-1(cosec θ) = π/2 - θ for θ in appropriate subintervals avoiding singular points.
• cosec^-1(sec θ) = π/2 - θ for θ ∈ [0, π] \ {π/2} (respecting principal values).
• sin^-1x = cos^-1[√(1 - x²)] for 0 ≤ x ≤ 1.
• sin^-1x = - cos^-1[√(1 - x²)] for -1 ≤ x < 0, with sign chosen to match principal />

Illustrations for Property Set 4

Example 1. Given cos^-1(-3/4) = π - sin^-1A. Find A.

Solution:

cos^-1(-3/4) = π - cos^-1(3/4) because cos^-1(-x) = π - cos^-1(x).

Consider a right triangle (or unit-circle construction) with adjacent side 3 and hypotenuse 4 so that sin of the acute angle is √(1 - (3/4)²) = √7/4.

Therefore A = √7/4.

Additional short examples and conversions:

• cos^-1(1/4) = sin^-1(√15/4).
• sin^-1(-1/2) = - sin^-1(1/2) = -π/6, and this equals - cos^-1(√3/2) appropriately interpreted in principal ranges.
• sin²(tan^-1(3/4)) = (3/5)² = 9/25 because tan θ = 3/4 gives sin θ = 3/5.
• sin^-1(sin 2π/3) = π/3 since 2π/3 is not in [-π/2, π/2] and the principal value maps to π/3.

Property Set 5 - Complementary sums

• sin^-1x + cos^-1x = π/2 for x ∈ [-1,1].
• tan^-1x + cot^-1x = π/2 for all real x (use principal branches).
• sec^-1x + cosec^-1x = π/2 for |x| ≥ 1.

Proof - sin^-1x + cos^-1x = π/2

Let sin^-1x = y, so x = sin y.

Then x = cos(π/2 - y).

Hence cos^-1x = π/2 - y.

Adding gives sin^-1x + cos^-1x = y + (π/2 - y) = π/2.

Illustrations

• sec^-1(4) + cosec^-1(4) = π/2.
• tan^-1(3) + cot^-1(3) = π/2.

Property Set 6 - Addition and subtraction formulas for arctangent

• tan^-1x + tan^-1y = tan^-1[(x + y)/(1 - xy)] when xy < 1 (adjust by ±π when necessary to get the principal />
• tan^-1x - tan^-1y = tan^-1[(x - y)/(1 + xy)] when xy > -1 (adjust by ±π as required).
• 2 tan^-1x = tan^-1[2x/(1 - x²)] for |x| < 1 (principal-value />

Proof - tan^-1x + tan^-1y formula

Let tan^-1x = α and tan^-1y = β, so x = tan α and y = tan β.

Then tan(α + β) = (tan α + tan β)/(1 - tan α tan β) = (x + y)/(1 - xy).

Therefore α + β = tan^-1[(x + y)/(1 - xy)] up to addition of kπ; choose k so that α + β lies in principal branch.

Examples using arctangent addition

Evaluation steps are shown without numbered "Step" labels but each algebraic/argument step is on a separate line.

Example. tan^-1(-1/2) + tan^-1(-1/3).

Compute (-1/2) + (-1/3) = -5/6.

Compute 1 - (-1/2)(-1/3) = 1 - 1/6 = 5/6.

So tan^-1(-1/2) + tan^-1(-1/3) = tan^-1((-5/6)/(5/6)) = tan^-1(-1) = -π/4.

Example. tan^-1(-2) + tan^-1(-3).

Sum numerator = -5 and denominator = 1 - 6 = -5.

So tan^-1(-2) + tan^-1(-3) = tan^-1(1) = π/4 (principal value).

Example (equation). tan^-12x + tan^-13x = π/4. Find x.

Use tan addition: tan(π/4) = 1 = (2x + 3x)/(1 - 6x²) = 5x/(1 - 6x²).

Therefore 5x = 1 - 6x².

Rearrange: 6x² + 5x - 1 = 0.

Solve quadratic: discriminant = 25 + 24 = 49.

Roots: x = (-5 ± 7)/12, i.e. x = 1/6 or x = -1.

Both satisfy the original equation after checking principal-value conditions.

Example. If tan^-14 + tan^-15 = cot^-1(λ), find λ.

Compute tan^-14 + tan^-15 = tan^-1[(4 + 5)/(1 - 20)] = tan^-1(9/(-19)) = tan^-1(-9/19).

So cot^-1(λ) = tan^-1(-9/19).

Use relation cot^-1(u) = tan^-1(1/u) with sign adjustments; here one finds λ = -19/9 by matching principal values.

Property Set 7 - Sum formulas for arcsine and arccosine

• sin^-1x + sin^-1y = sin^-1[x√(1 - y²) + y√(1 - x²)] provided the right-hand side expression lies in [-1,1] and principal ranges are used.
• cos^-1x + cos^-1y = cos^-1[xy - √(1 - x²)√(1 - y²)] under appropriate conditions on x,y.

Illustration for sum of arcsines

Example. sin^-1(4/5) + sin^-1(7/25) = sin^-1(A). Find A.

Compute √(1 - (7/25)²) and √(1 - (4/5)²).

Evaluate A = (4/5)√(1 - (7/25)²) + (7/25)√(1 - (4/5)²) = 117/125.

So sin^-1(4/5) + sin^-1(7/25) = sin^-1(117/125).

Example (verification of complementary sum). Prove sin^-1(63/65) + sin^-1(16/65) = π/2.

Use sin-cos complement: sin^-1(63/65) = cos^-1(16/65).

Therefore sin^-1(63/65) + sin^-1(16/65) = cos^-1(16/65) + sin^-1(16/65) = π/2.

Property Set 8 - Behaviour of f(f(x)) graphs and periodic folding

• sin^-1(sin x) equals x if x ∈ [-π/2, π/2], and takes different folded values outside this interval according to periodicity and principal branch (for example π - x, -π - x, etc.).
• cos^-1(cos x) equals x if x ∈ [0, π], and takes appropriate folded values outside that interval (for example 2π - x in some intervals).
• tan^-1(tan x) equals x for x ∈ (-π/2, π/2) and equals x - π or x + π when x lies outside that principal interval, according to periodic continuation by π.

Typical evaluations using these rules:

• sin^-1(sin 2π/3) = π/3.
• cos^-1(cos 13π/6) = π/6.
• sin^-1(sin 4) = π - 4 (because 4 ≈ 1.273π lies in [π/2, 3π/2], fold back to principal range).
• sin^-1(sin 6) = 6 - 2π.
• tan^-1(tan 3) = 3 - π (since 3 ∉ (-π/2, π/2)).

Property Set 9 - Double-angle connection for arctangent

Relation: 2 tan^-1x = sin^-1[2x/(1 + x²)] for |x| < 1 (principal branches chosen />

Proof

Let tan^-1x = y, so x = tan y.

Then 2 tan^-1x = 2y.

Compute 2x/(1 + x²) = (2 tan y)/(1 + tan²y) = sin 2y.

So sin^-1[2x/(1 + x²)] = sin^-1(sin 2y) = 2y.

Thus 2 tan^-1x = sin^-1[2x/(1 + x²)].

• Also 2 tan^-1x = cos^-1[(1 - x²)/(1 + x²)] for x ≥ 0, with appropriate principal-value interpretation.
• And 2 tan^-1x = tan^-1[2x/(1 - x²)] when -1 < x />< 1 (again choosing the correct branch so the equality />

Further illustrative figures and solved diagrams

Concluding remarks

When working with inverse trigonometric functions always:

• Keep principal branches (ranges) in mind; many apparent algebraic equalities require choosing the correct branch and sometimes adding or subtracting π (or multiples) to match principal values.
• Check domains carefully before applying reciprocal or composition relations (for example, arcsec and arccosec require |x| ≥ 1).
• Use geometric (triangle or unit-circle) constructions to evaluate or simplify expressions like sin^-1 and cos^-1 when a ratio is given.
• When using addition/subtraction formulas for arctangent, always verify the sign/branch of the resulting angle so the principal value is obtained.

This document collects the standard properties, identities and typical illustrative examples required for a clear understanding of inverse trigonometric functions and their use in problem solving. Apply the principal-value conventions consistently to avoid sign or π-shift errors.