The value of $\left(\sec A + \tan A\right) (1 - \sin A) $ is: | Mathematics MCQ | Examoogle

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The value of $\left(\sec A + \tan A\right) (1 - \sin A)$ is:

$\tan A$

$\cot A$

$\sin A$

$\cos A$

Correct Answer

📐 MA • Math Algebraic SimplificationMathematicsPrevious Year Paper 2024

Option D

$\cos A$

Quick Summary:

Substitute a simple value like $A = 0°$ . Then $(\sec 0 + \tan 0)(1 - \sin 0) = (1 + 0)(1 - 0) = 1$ . Checking the options at $A = 0°$ : $\tan 0 = 0$ , $\cot 0$ is undefined, $\sin 0 = 0$ , $\cos 0 = 1$ . Only $\cos A$ matches.

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📐MAMath SolutionAlgebraic Simplification

📋 Given

An algebraic expression in trigonometry: (sec A + tan A)(1 - sin A)

🔢 Formula Used

sec⁡A=1cos⁡A,tan⁡A=sin⁡Acos⁡A,sin⁡2A+cos⁡2A=1\sec A = \frac{1}{\cos A}, \tan A = \frac{\sin A}{\cos A}, \sin^2 A + \cos^2 A = 1secA=cosA1​,tanA=cosAsinA​,sin2A+cos2A=1

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Many students get stuck trying to expand the expression directly without converting secant and tangent into sine and cosine first.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Convert terms to sine and cosine

Express $\sec A$ as $\frac{1}{\cos A}$ and $\tan A$ as $\frac{\sin A}{\cos A}$ .

(1cos⁡A+sin⁡Acos⁡A)(1−sin⁡A)(\frac{1}{\cos A} + \frac{\sin A}{\cos A})(1 - \sin A)(cosA1​+cosAsinA​)(1−sinA)

Simplify the sum

Combine the fractions inside the first parenthesis since they share a common denominator.

(1+sin⁡Acos⁡A)(1−sin⁡A)(\frac{1 + \sin A}{\cos A})(1 - \sin A)(cosA1+sinA​)(1−sinA)

Multiply the expressions

Perform the multiplication in the numerator using the difference of squares identity $(1 + \sin A)(1 - \sin A) = 1 - \sin^2 A$ .

1−sin⁡2Acos⁡A\frac{1 - \sin^2 A}{\cos A}cosA1−sin2A​

Use Pythagorean identity

Substitute $1 - \sin^2 A$ with $\cos^2 A$ and simplify the fraction.

cos⁡2Acos⁡A=cos⁡A\frac{\cos^2 A}{\cos A} = \cos AcosAcos2A​=cosA

✅

D is correct because the trigonometric simplification of the given expression results in cos A.

Core Concepts Used

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Trigonometric Identities Algebraic Simplification Pythagorean Theorem

💡 EXAM TIP

This type of simplification frequently appears in SSC and Railway exams. Mastering identities like $\sin^2 \theta + \cos^2 \theta = 1$ is crucial for solving such problems quickly.

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