JEE Main Maths Trigonometry 2027: Identities, Compound Angles & Inverse Functions

Trigonometry is the last chapter of the JEE Main Maths syllabus, and in many ways it is the most connective, since it shows up inside Coordinate Geometry, Vectors and Complex Numbers as well. A clear picture of the unit circle makes every identity in this chapter feel inevitable rather than memorised.

Every trigonometric identity is really a statement about the same circle, seen from a different angle.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Trigonometrical identities and trigonometrical functions
• Inverse trigonometrical functions and their properties

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Master These Topics

1. The Unit Circle and Core Identities

$$\sin^2\theta + \cos^2\theta = 1 \qquad 1+\tan^2\theta = \sec^2\theta \qquad 1+\cot^2\theta = \csc^2\theta$$

2. Compound and Multiple Angle Formulas

$$\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B \qquad \cos(A\pm B) = \cos A\cos B \mp \sin A\sin B$$

$$\tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$

Worked example: $\cos 75° = \cos(45°+30°) = \cos45°\cos30° - \sin45°\sin30° = \dfrac{\sqrt6-\sqrt2}{4}$.

Double angle formulas: $\sin2\theta = 2\sin\theta\cos\theta$, and $\cos2\theta$ has three equivalent forms:

$$\cos2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

Worked example: If $\sin\theta = \tfrac{3}{5}$ with $\theta$ in the first quadrant, then $\cos\theta = \tfrac{4}{5}$, so $\sin2\theta = 2 \cdot \tfrac{3}{5}\cdot\tfrac{4}{5} = \tfrac{24}{25}$ and $\cos2\theta = 1 - 2\cdot\tfrac{9}{25} = \tfrac{7}{25}$.

Trap: There are three forms of $\cos2\theta$. Pick the one that matches what is given (in terms of $\sin\theta$ alone, $\cos\theta$ alone, or both), rather than always expanding fully.

3. Inverse Trigonometric Functions

Each inverse function has a restricted range, called its principal value branch, so that it is a genuine function:

Key properties:

$$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \qquad \tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \; (xy \lt 1)$$

Worked example: $\sin^{-1}\tfrac{1}{2} + \cos^{-1}\tfrac{1}{2} = \tfrac{\pi}{6}+\tfrac{\pi}{3} = \tfrac{\pi}{2}$, confirming the identity.

Worked example (a classic result): $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi$. Using the addition formula on the first two terms: $\tan^{-1}1+\tan^{-1}2 = \pi + \tan^{-1}\left(\dfrac{1+2}{1-2}\right) = \pi - \tan^{-1}3$ (adjusted for the quadrant, since $xy = 2 \gt 1$), which added to $\tan^{-1}3$ gives $\pi$.

Trap: The addition formula for $\tan^{-1}x+\tan^{-1}y$ needs an adjustment of $\pm\pi$ when $xy \gt 1$, since the plain formula can land outside the principal range.

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Common Traps to Avoid

• Forgetting the restricted range of an inverse trigonometric function and accepting an out-of-range answer.
• Using the addition formula for $\tan^{-1}x+\tan^{-1}y$ without the correction term when $xy \gt 1$.
• Picking the wrong version of the triple $\cos2\theta$ formula for the given information.
• Mixing up compound angle signs, such as writing $\cos(A+B)$ with a $+$ between the product terms.

60-Second Revision Sheet

• $\sin^2\theta+\cos^2\theta=1$; $1+\tan^2\theta=\sec^2\theta$
• $\sin(A\pm B) = \sin A\cos B\pm\cos A\sin B$; $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
• $\cos2\theta = 1-2\sin^2\theta = 2\cos^2\theta-1$
• $\sin^{-1}x+\cos^{-1}x = \pi/2$; ranges: $\sin^{-1}\to[-\pi/2,\pi/2]$, $\cos^{-1}\to[0,\pi]$, $\tan^{-1}\to(-\pi/2,\pi/2)$

Your Study Plan

1. Day 1: basic identities and the unit circle.
2. Day 2: compound and multiple angle formulas.
3. Day 3: inverse trigonometric functions, ranges and properties.
4. Day 4: mixed identity and inverse trig problems, timed set.

Practice Trigonometry Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Statistics & Probability
• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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Frequently Asked Questions