medium difficulty5 min

Using trig identities to simplify expressions

Using trigonometric identities to simplify expressions means replacing one trig form with another so the expression becomes shorter or easier to solve.

In one line: Spot the full pattern, then substitute the identity like a puzzle piece.

Practice This Concept

Understand it

Many trigonometry questions ask students to simplify an expression instead of calculating a final numeric value. The identities sin squared A plus cos squared A equals 1, 1 plus tan squared A equals sec squared A, and 1 plus cot squared A equals cosec squared A are especially useful. The goal is to recognize which identity fits the expression best.

Example

Expression: 1 plus tan squared A. Using the identity, it becomes sec squared A.

Formula and rule

Use the standard identities: sin² A + cos² A = 1, 1 + tan² A = sec² A, and 1 + cot² A = cosec² A.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1If an expression is sec squared A minus tan squared A, rewrite sec squared A as 1 plus tan squared A.
2Then the expression becomes 1.

Method you can reuse

1Inspect the expression.
2Identify the complete identity pattern.
3Replace the matching part.
4Simplify carefully.
5Check the final form.

Visual tip

A small right-triangle sketch can help when the expression comes from a side ratio, but many simplifications can be done directly from identities.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How do you simplify 1 + cot squared A using a standard identity?

Think first, then show answer

It simplifies directly to cosec squared A. The full identity is 1 plus cot squared A equals cosec squared A.

Common wrong answer

A common wrong answer is to change only part of the identity and ignore the square terms. That breaks the algebra.

Don't mix this up

Students often try to expand every term instead of spotting the matching identity first.

Exam tip

Look for a complete identity pattern before doing any algebraic steps.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

CBSE often asks students to simplify trig expressions, prove a left-hand side equals a right-hand side, or complete a missing term in an identity.

1
Enough for 1 mark
Using trigonometric identities to simplify expressions means replacing one trig form with another so the expression becomes shorter or easier to solve.
2
Add for 2 marks
Example: Expression: 1 plus tan squared A. Using the identity, it becomes sec squared A.
3
Add for 3 marks
The identities sin squared A plus cos squared A equals 1, 1 plus tan squared A equals sec squared A, and 1 plus cot squared A equals cosec squared A are especially useful.

MCQ Quiz

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