Which of the following is equal to $\sec^2 \theta - 1$? | Mathematics MCQ | Examoogle

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Which of the following is equal to $\sec^2 \theta - 1$ ?

$\sin^2 \theta$

$\cos^2 \theta$

$\tan^2 \theta$

$\cot^2 \theta$

Correct Answer

📐 MA • Math Direct FormulaMathematicsTrigonometry

Option C

$\tan^2 \theta$

Quick Summary:

Remember the primary identity $\sin^2 \theta + \cos^2 \theta = 1$ . Dividing by $\cos^2 \theta$ gives $1 + \tan^2 \theta = \sec^2 \theta$ , so directly $\sec^2 \theta - 1 = \tan^2 \theta$ .

📐MAMath SolutionDirect Formula

📋 Given

The expression provided is the trigonometric identity component sec² θ - 1.

🔢 Formula Used

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ

⚡ Exam Hall Shortcut / Speed Trick

Remember the primary identity $\sin^2 \theta + \cos^2 \theta = 1$ . Dividing by $\cos^2 \theta$ gives $1 + \tan^2 \theta = \sec^2 \theta$ , so directly $\sec^2 \theta - 1 = \tan^2 \theta$ .

⚠️ Common Student Trap / Pitfall

Students often confuse the identity with $\sec^2 \theta + 1$ or incorrectly rearrange it to $1 - \sec^2 \theta = -\tan^2 \theta$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Recall Fundamental Trigonometric Identity

Start with the Pythagorean identity relating secant and tangent functions: $1 + \tan^2 \theta = \sec^2 \theta$ .

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ

Rearrange the Equation

Subtract 1 from both sides of the identity to isolate the expression requested in the question.

tan⁡2θ=sec⁡2θ−1\tan^2 \theta = \sec^2 \theta - 1tan2θ=sec2θ−1

Match with Options

The derived result $\tan^2 \theta$ matches the content provided in option C.

Result=tan⁡2θ\text{Result} = \tan^2 \thetaResult=tan2θ

✅

C is correct because the trigonometric identity $1 + \tan^2 \theta = \sec^2 \theta$ directly rearranges to $\sec^2 \theta - 1 = \tan^2 \theta$ .

Core Concepts Used

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Trigonometric Identities Pythagorean Theorem Rearrangement of algebraic expressions

💡 EXAM TIP

This concept is the foundation for trigonometric integration and substitution techniques in Calculus, specifically when solving integrals containing the term $\sqrt{x^2 + a^2}$ .

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