The value of $\left[\frac{\cos 50^{\circ}}{1 + \sin 50^{\circ}}\right] + \left[\frac{1 + \sin 50^{\circ}}{\cos 50^{\circ}}\right]$ is:

The correct answer is Option B: $2 \sec 50^{\circ}$. Use the identity $\frac{\cos \theta}{1+\sin \theta} = \tan(45^\circ - \frac{\theta}{2})$. Let $x = 50^\circ$. The expression is $\tan(45^\circ - 25^\circ) + \cot(45^\circ - 25^\circ) = \tan 20^\circ + \cot 20^\circ = \frac{2}{\sin 40^\circ} = 2 \operatorname{cosec} 40^\circ$. Since options use 50°, note $\operatorname{cosec} 40^\circ = \sec 50^\circ$. Step-by-Step Solution & Concepts: Shortcut Trick: Use the identity $\frac{\cos \theta}{1+\sin \theta} = \tan(45^\circ - \frac{\theta}{2})$. Let $x = 50^\circ$. The expression is $\tan(45^\circ - 25^\circ) + \cot(45^\circ - 25^\circ) = \tan 20^\circ + \cot 20^\circ = \frac{2}{\sin 40^\circ} = 2 \operatorname{cosec} 40^\circ$. Since options use 50°, note $\operatorname{cosec} 40^\circ = \sec 50^\circ$. Formula Used: $$ \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab} = \frac{(a+b)^2}{ab} - 2 $$ and half-angle/complementary relations Given: Expression to evaluate: [cos 50° / (1 + sin 50°)] + [(1 + sin 50°) / cos 50°] Steps: Combine Fractions: Let $a = \cos 50^\circ$ and $b = 1 + \sin 50^\circ$. The expression is $\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$.. Simplify Numerator: Expand $(1 + \sin 50^\circ)^2 = 1 + 2\sin 50^\circ + \sin^2 50^\circ$. Using $\cos^2 50^\circ + \sin^2 50^\circ = 1$, the numerator becomes $1 + 1 + 2\sin 50^\circ = 2(1 + \sin 50^\circ)$.. Final Calculation: Divide the numerator by the denominator: $\frac{2(1 + \sin 50^\circ)}{\cos 50^\circ(1 + \sin 50^\circ)} = \frac{2}{\cos 50^\circ} = 2 \sec 50^\circ$. Answer: B is correct because the algebraic simplification of the given trigonometric expression results in 2 sec 50°. Exam Tip: Always look for the $(1+\sin \theta)$ pattern in denominators; it often simplifies beautifully with $\cos \theta$ using the half-angle substitution $1+\sin \theta = (\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^2$.

The value of $\left[\frac{\cos 50^{\circ}}{1 + \sin 50^{\circ}... | Mathematics MCQ

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MathematicsPrevious Year Paper 2024

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The value of $\left[\frac{\cos 50°}{1 + \sin 50°}\right] + \left[\frac{1 + \sin 50°}{\cos 50°}\right]$ is:

$2 \tan 50°$

$2 \sec 50°$

$2 \cot 50°$

$2 \operatorname{cosec} 50°$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option B

$2 \sec 50°$

Quick Summary:

Use the identity $\frac{\cos \theta}{1+\sin \theta} = \tan(45°- \frac{\theta}{2})$ . Let $x = 50°$ . The expression is $\tan(45°- 25°) + \cot(45°- 25°) = \tan 20°+ \cot 20°= \frac{2}{\sin 40° = 2 \operatorname{cosec} 40°$ . Since options use 50°, note $\operatorname{cosec} 40°= \sec 50°$ .

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📐MAMath SolutionDirect Formula

📋 Given

Expression to evaluate: [cos 50° / (1 + sin 50°)] + [(1 + sin 50°) / cos 50°]

🔢 Formula Used

ab+ba=a2+b2ab=(a+b)2−2abab=(a+b)2ab−2\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab} = \frac{(a+b)^2}{ab} - 2ba​+ab​=aba2+b2​=ab(a+b)2−2ab​=ab(a+b)2​−2 and half-angle/complementary relations

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Many students try to expand the denominators incorrectly. The simplest path is to combine the fractions using a common denominator.

🔢 Step-by-Step Solution

Combine Fractions

Let $a = \cos 50°$ and $b = 1 + \sin 50°$ . The expression is $\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$ .

cos⁡250°+(1+sin⁡50°)2cos⁡50°(1+sin⁡50°)\frac{\cos^2 50°+ (1 + \sin 50°)^2}{\cos 50°(1 + \sin 50°)}cos50°(1+sin50°)cos250°+(1+sin50°)2​

Simplify Numerator

Expand $(1 + \sin 50°)^2 = 1 + 2\sin 50°+ \sin^2 50°$ . Using $\cos^2 50°+ \sin^2 50°= 1$ , the numerator becomes $1 + 1 + 2\sin 50°= 2(1 + \sin 50°)$ .

2+2sin⁡50°=2(1+sin⁡50°)2 + 2\sin 50°= 2(1 + \sin 50°)2+2sin50°=2(1+sin50°)

Final Calculation

Divide the numerator by the denominator: $\frac{2(1 + \sin 50°)}{\cos 50°(1 + \sin 50°)} = \frac{2}{\cos 50° = 2 \sec 50°$ .

2sec⁡50°2 \sec 50°2sec50°

✅

B is correct because the algebraic simplification of the given trigonometric expression results in 2 sec 50°.

Core Concepts Used

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Trigonometric Identities Fractional Addition Pythagorean Identity

💡 EXAM TIP

Always look for the $(1+\sin \theta)$ pattern in denominators; it often simplifies beautifully with $\cos \theta$ using the half-angle substitution $1+\sin \theta = (\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^2$ .

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