Trigonometric identities master list — with derivation connections

Question

List all essential trigonometric identities in a logical order showing how each one derives from the previous ones. Which identities are most important for CBSE and JEE?

(CBSE 10/11 + JEE Main — foundation for all trig problems)

Solution — Step by Step

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

Divide by $\cos^2\theta$ : $\tan^2\theta + 1 = \sec^2\theta$

Divide by $\sin^2\theta$ : $1 + \cot^2\theta = \csc^2\theta$

These three come from Pythagoras applied to the unit circle. Everything else builds on them.

\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

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

These are derived from rotation matrices or geometric constructions. They generate all the identities below.

\sin 2A = 2\sin A \cos A

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

\tan 2A = \frac{2\tan A}{1 - \tan^2 A}

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

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

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

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

flowchart TD
    A["sin²θ + cos²θ = 1"] --> B["Divide by cos²θ"]
    A --> C["Divide by sin²θ"]
    B --> D["tan²θ + 1 = sec²θ"]
    C --> E["1 + cot²θ = csc²θ"]
    A --> F["Compound Angle Formulas"]
    F --> G["Set B = A"]
    G --> H["Double Angle Formulas"]
    H --> I["Replace A with A/2"]
    I --> J["Half Angle Formulas"]
    F --> K["Add/Subtract pairs"]
    K --> L["Product-to-Sum Formulas"]
    L --> M["Reverse direction"]
    M --> N["Sum-to-Product Formulas"]

Why This Works

All trigonometric identities are ultimately consequences of the definition of sin and cos on the unit circle, combined with the Pythagorean theorem. You do not need to memorise dozens of disconnected formulas — understand the derivation tree, and you can re-derive any identity in 30 seconds.

The compound angle formulas are the single most important identities after Pythagoras. Everything — double angle, half angle, product-to-sum, sum-to-product — follows from them by substitution.

Alternative Method

For JEE, a powerful alternative to sum-to-product formulas is the auxiliary angle method:

$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$

where $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \frac{b}{a}$ .

This converts a sum of sin and cos into a single sinusoidal function — extremely useful for finding maximum/minimum values and solving equations.

For CBSE 10, you need only the three Pythagorean identities. For CBSE 11 and JEE, you need all of them. The most-tested in JEE Main: compound angle formulas and double angle formulas. If you know these cold, you can handle 90% of trig problems.

Common Mistake

The most dangerous error: writing $\cos(A + B) = \cos A + \cos B$ . The cosine of a sum is NOT the sum of cosines. The correct formula has products: $\cos A \cos B - \sin A \sin B$ . This error cascades into every problem that uses compound angles. Whenever you see $\cos(A + B)$ , mentally flag that it requires the full expansion.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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