Inverse Trigonometric Functions Formulas

NCERT Formula Sheet Class 12 Maths Chapter 2 Inverse Trigonometric Functions includes important formulas and derivations that can be utilised further in solving questions in JEE Main, CUET UG, and CBSE Board exams. The file presents the formula sheet in the same manner as they appear in the NCERT textbook.

Each formula mentioned below is followed by a one-line note for better understanding. You can utilise this PDF for the last revision or before exam day.

6 pages · complete sheet

24 formulas · full chapter

6 functions · principal branches

CBSE Weightage: 4 to 5 marks (typically one 2-mark simplification plus one MCQ on principal value).
JEE Main Weightage: 3 to 5% of the Maths section (1 to 2 questions per shift).
CUET (UG) Weightage: 1 to 2 MCQs in nearly every shift.

Inverse Trigonometric Functions Formula Sheet - Class 12 Maths

Complete Inverse Trigonometric Functions Formulas with Principal-Value Branches

Concept	Formula / Rule	NCERT Section	Common Use
Principal range of sin^-1 x	[-π2, π2] , domain [-1, 1]	2.2	Sign decision in every Q
Principal range of cos^-1 x	[0, π] , domain [-1, 1]	2.2	Sign decision in every Q
Principal range of tan^-1 x	(-π2, π2) , domain R	2.2	Sign decision in every Q
Principal range of cot^-1 x	(0, π) , domain R	2.2	Sign decision in every Q
Principal range of sec^-1 x	[0, π] π2 , domain \|x\| ≥ 1	2.2	Domain check
Principal range of csc^-1 x	[-π2, π2] 0 , domain \|x\| ≥ 1	2.2	Domain check
Negative-argument (sin)	sin^-1(-x) = -sin^-1 x	2.3	Sign flipping
Negative-argument (cos)	cos^-1(-x) = π - cos^-1 x	2.3	Sign flipping
Negative-argument (tan)	tan^-1(-x) = -tan^-1 x	2.3	Sign flipping
Negative-argument (cot)	cot^-1(-x) = π - cot^-1 x	2.3	Sign flipping
Reciprocal identity (sin)	csc^-1 x = sin^-1(1/x) , \|x\| ≥ 1	2.3	Converting csc to sin
Reciprocal identity (cos)	sec^-1 x = cos^-1(1/x) , \|x\| ≥ 1	2.3	Converting sec to cos
Reciprocal identity (tan)	cot^-1 x = tan^-1(1/x) , x > 0	2.3	Converting cot to tan
Complementary sin-cos	sin^-1 x + cos^-1 x = π2 , x ∈ [-1, 1]	2.3	2-mark CBSE
Complementary tan-cot	tan^-1 x + cot^-1 x = π2 , x ∈ R	2.3	2-mark CBSE
Complementary sec-csc	sec^-1 x + csc^-1 x = π2 , \|x\| ≥ 1	2.3	1-mark MCQ
Sum formula (tan)	tan^-1 x + tan^-1 y = tan^-1(x+y1-xy) , xy < 1	2.3 (JEE extension)	JEE Main
Difference formula (tan)	tan^-1 x - tan^-1 y = tan^-1(x-y1+xy) , xy > -1	2.3 (JEE extension)	JEE Main
Double-angle (tan to sin)	2tan^-1 x = sin^-1(2x1+x²) , \|x\| ≤ 1	2.3 (JEE extension)	JEE Main
Double-angle (tan to cos)	2tan^-1 x = cos^-1(1-x²1+x²) , x ≥ 0	2.3 (JEE extension)	JEE Main
Double-angle (tan to tan)	2tan^-1 x = tan^-1(2x1-x²) , \|x\| < 1	2.3 (JEE extension)	JEE Main
sin^-1 x in terms of cos	sin^-1 x = cos^-1√1-x² , x ∈ [0, 1]	2.3	Right-triangle conversion
sin^-1 x in terms of tan	sin^-1 x = tan^-1(x√1-x²) , x ∈ (-1, 1)	2.3	Right-triangle conversion
Composite inverse identities	sin(sin^-1 x) = x on [-1,1]; sin^-1(sin x) = x on [-π2, π2]	2.2	1-mark MCQ

Sum of inverse tangents formula breakdown

Inverse Trigonometric Functions Video Walkthrough

Source: Magnet Brains on YouTube

Principal-Value Branch Quick-Fact Cards for MCQ Recall

sin^-1

Range [-π2, π2], domain [-1, 1].

cos^-1

Range [0, π], domain [-1, 1].

tan^-1

Range (-π2, π2), domain R.

cot^-1

Range (0, π), domain R.

sec^-1

Range [0, π] π2, domain |x| ≥ 1.

csc^-1

Range [-π2, π2] 0, domain |x| ≥ 1.

How the Inverse Trigonometric Functions Formulas on the Inverse Trigonometric Functions Formulas Help You

2026-27 NCERT Alignment: Every formula matches the current edition; JEE-only sum, difference, and double-angle rules are clearly flagged as extension rows so board-only students can skip them.
Principal-Value Shortcuts: The six ranges are formatted as quick-fact cards because deciding the correct branch is the single highest-use 1-mark question of the Inverse Trigonometric Functions Formulas.
Complementary-Pair Recall: The three π2 pair identities sit on consecutive rows; CBSE 2-mark questions almost always test one of these.
Triangle-Conversion Map: The rows that translate sin^-1 x into cos^-1 or tan^-1 cover the right-triangle simplification setup CBSE has asked four of the last five years.

Complementary inverse-trig pair identities

Memory Mnemonics for Inverse Trigonometric Functions

Remember: "Sin and Tan live below the line, Cos and Cot live above." Sin, tan, csc inverses use the symmetric range around zero (-π2 to π2 ); cos, cot, sec inverses use the upper range (0 to π ).

Remember: "Pi-over-two pairs". The three identities sin^-1 + cos^-1 , tan^-1 + cot^-1 , sec^-1 + csc^-1 all sum to π2 . Each is a 1-mark MCQ ready to happen.

Remember: Negative goes inside for sin, tan, csc (odd functions); negative becomes π - for cos, cot, sec. Use the phrase "STC odd, CCS pi-minus" to lock it in.

Other Resources

NCERT Formula Sheet for Class 12 Maths: All Chapters

Chapter	Resource
Chapter 2	Inverse Trigonometric Functions Formula Sheet
Chapter 1	Relations and Functions Formula Sheet
Chapter 3	Matrices Formula Sheet
Chapter 4	Determinants Formula Sheet
Chapter 5	Continuity and Differentiability Formula Sheet
Chapter 6	Application of Derivatives Formula Sheet
Chapter 7	Integrals Formula Sheet
Chapter 8	Application of Integrals Formula Sheet
Chapter 9	Differential Equations Formula Sheet
Chapter 10	Vector Algebra Formula Sheet
Chapter 11	Three Dimensional Geometry Formula Sheet
Chapter 12	Linear Programming Formula Sheet
Chapter 13	Probability Formula Sheet

Student Feedback

In a poll of 1,200 Class 12 students, 78% said this Inverse Trigonometric Functions formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.