medium difficulty5 min

Identity 1 + tan^2 A = sec^2 A

The identity 1 plus tan squared A equals sec squared A is a standard trigonometric identity used for simplification and verification.

In one line: Tangent to secant is a square-step identity: square first, then add one.

Practice This Concept

Understand it

This identity is derived from the basic relation sin squared A plus cos squared A equals 1. If the whole equation is divided by cos squared A, the result becomes 1 plus tan squared A equals sec squared A. It is very useful when a problem contains tangent and secant together.

Formula and rule

1 + tan² A = sec² A.

This comes from dividing sin² A + cos² A = 1 by cos² A.

See it worked out

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

Worked example

If tan A = 1/2, then sec squared A = 1 + 1/4 = 5/4, so sec A = root 5 over 2 for an acute angle.

Method you can reuse

1Identify tan A.
2Square it.
3Add 1.
4Take the square root only if sec A, not sec squared A, is required.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If tan A = 3, what is sec squared A according to the identity?

Think first, then show answer

sec squared A = 1 + tan squared A = 1 + 9 = 10. The square on tangent must be used before adding 1.

Common wrong answer

A common wrong answer is to add 1 to tan A without squaring. That gives the wrong result and breaks the identity.

Don't mix this up

Students often write 1 plus tan A equals sec A, but the squares are required on both trig terms.

Exam tip

Whenever tangent is given and secant is needed, use this identity directly instead of building a triangle again.

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 may ask for a missing secant value, or ask you to simplify an expression that contains 1 plus tan squared A.

1
Enough for 1 mark
The identity 1 plus tan squared A equals sec squared A is a standard trigonometric identity used for simplification and verification.
2
Add for 2 marks
Example: If tan A = 2, then sec squared A = 1 + 4 = 5, so sec A = root 5 for an acute angle.
3
Add for 3 marks
This identity is derived from the basic relation sin squared A plus cos squared A equals 1.

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