Find the value of the given expression.$\frac{\cos A}{1 - \ta... | Mathematics MCQ | Examoogle

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Find the value of the given expression. $\frac{\cos A}{1 - \tan A} + \frac{\sin^2 A}{\sin A - \cos A} + 2\sin A - \cos A$

$3\sin A$

$\sin A$

$3\cos A$

$\cos A$

Correct Answer

📐 MA • Math Algebraic SimplificationMathematicsTrigonometry

Option A

$3\sin A$

Quick Summary:

Substitute $A = 90°$ is undefined due to $\tan A$ , so choose $A = 45°$ . The expression yields: (1/sqrt(2) / 0) - (0.5 / 0)... wait, check $A = 0°$ : $1/1 + 0/(-1) + 0 - 1 = 1 - 1 = 0$ . Since $\sin(0) = 0$ , the result $3\sin A$ matches this.

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📐MAMath SolutionAlgebraic Simplification

📋 Given

An expression containing trigonometric terms: (cos A / (1 - tan A)) + (sin² A / (sin A - cos A)) + 2 sin A - cos A

🔢 Formula Used

tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}tanA=cosAsinA​, sin⁡2A−cos⁡2A=(sin⁡A−cos⁡A)(sin⁡A+cos⁡A)\sin^2 A - \cos^2 A = (\sin A - \cos A)(\sin A + \cos A)sin2A−cos2A=(sinA−cosA)(sinA+cosA)

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often struggle with the sign convention in the second term when switching the denominator from $(\sin A - \cos A)$ to $(\cos A - \sin A)$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify the first term

Substitute $\tan A = \frac{\sin A}{\cos A}$ in the first term: $\frac{\cos A}{1 - \frac{\sin A}{\cos A}} = \frac{\cos^2 A}{\cos A - \sin A}$

cos⁡2Acos⁡A−sin⁡A\frac{\cos^2 A}{\cos A - \sin A}cosA−sinAcos2A​

Combine with the second term

Rewrite the expression by making denominators consistent: $\frac{\cos^2 A}{\cos A - \sin A} - \frac{\sin^2 A}{\cos A - \sin A} = \frac{\cos^2 A - \sin^2 A}{\cos A - \sin A}$

(cos⁡A−sin⁡A)(cos⁡A+sin⁡A)cos⁡A−sin⁡A=cos⁡A+sin⁡A\frac{(\cos A - \sin A)(\cos A + \sin A)}{\cos A - \sin A} = \cos A + \sin AcosA−sinA(cosA−sinA)(cosA+sinA)​=cosA+sinA

Simplify the full expression

Now add the remaining parts of the original expression: $(\cos A + \sin A) + 2\sin A - \cos A$ . Combining like terms: $\cos A - \cos A + \sin A + 2\sin A = 3\sin A$

3sin⁡A3\sin A3sinA

✅

A is correct because the entire expression simplifies algebraically to the term 3 sin A.

Core Concepts Used

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trigonometric identity algebraic simplification fraction subtraction

💡 EXAM TIP

Trigonometric simplification often relies on converting all functions to sine and cosine before combining terms.

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