C
CraftExam
high importancemedium8 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.

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

1 + tan^2 A = sec^2 A. This comes from dividing sin^2 A + cos^2 A = 1 by cos^2 A.

Memory hook

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

Examples and method

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 to apply

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

Diagram support

A right triangle helps students connect tangent with secant through the same angle A, especially when using the identity for a calculation.

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.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

The identity 1 plus tan squared A equals sec squared A is a standard trigonometric identity used for simplification and verification. 1 + tan^2 A = sec^2 A. This comes from dividing sin^2 A + cos^2 A = 1 by cos^2 A. If tan A = 2, then sec squared A = 1 + 4 = 5, so sec A = root 5 for an acute angle.

3-mark answer

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. 1 + tan^2 A = sec^2 A. This comes from dividing sin^2 A + cos^2 A = 1 by cos^2 A. 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. CBSE may ask for a missing secant value, or ask you to simplify an expression that contains 1 plus tan squared A. A common wrong answer is to add 1 to tan A without squaring. That gives the wrong result and breaks the identity.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept