The value of $\sin^2 \theta + \cos^2 \theta - (\sec^2 \theta -... | Mathematics MCQ | Examoogle

MathematicsPrevious Year Paper 2024

The value of $\sin^2 \theta + \cos^2 \theta - (\sec^2 \theta - \tan^2 \theta) + \tan \theta \cos \theta + \sin \theta$ is ________.

$\sec^2 \theta$

$2\sin \theta$

$-1$

$0$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option B

$2\sin \theta$

Quick Summary:

Use the fundamental identities $\sin^2 \theta + \cos^2 \theta = 1$ and $\sec^2 \theta - \tan^2 \theta = 1$ to simplify the first two terms immediately to $1 - 1 = 0$ , then simplify $\tan \theta \cos \theta$ to $\sin \theta$ .

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📐MAMath SolutionDirect Formula

📋 Given

An expression containing basic trigonometric identities: $\sin^2 \theta + \cos^2 \theta - (\sec^2 \theta - \tan^2 \theta) + \tan \theta \cos \theta + \sin \theta$ .

🔢 Formula Used

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1, sec⁡2θ−tan⁡2θ=1\sec^2 \theta - \tan^2 \theta = 1sec2θ−tan2θ=1, tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often struggle with the sign of the parenthesis group $-( \sec^2 \theta - \tan^2 \theta )$ or misremember the identity $\tan^2 \theta - \sec^2 \theta = 1$ (which is actually $-1$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify using fundamental identities

Apply the identities $\sin^2 \theta + \cos^2 \theta = 1$ and $\sec^2 \theta - \tan^2 \theta = 1$ to the first part of the expression.

1−(1)+tan⁡θcos⁡θ+sin⁡θ1 - (1) + \tan \theta \cos \theta + \sin \theta1−(1)+tanθcosθ+sinθ

Simplify the product term

Use the quotient identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to rewrite $\tan \theta \cos \theta$ .

sin⁡θcos⁡θ⋅cos⁡θ=sin⁡θ\frac{\sin \theta}{\cos \theta} \cdot \cos \theta = \sin \thetacosθsinθ​⋅cosθ=sinθ

Final combination

Substitute the simplified product back into the expression to get the final result.

0+sin⁡θ+sin⁡θ=2sin⁡θ0 + \sin \theta + \sin \theta = 2 \sin \theta0+sinθ+sinθ=2sinθ

✅

B is correct because the expression simplifies to $2 \sin \theta$ .

Core Concepts Used

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Trigonometric Identities Simplification Substitution

💡 EXAM TIP

These fundamental trigonometric identities are the foundation for solving calculus derivatives and integration problems involving trigonometric functions.

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