Simplify: $\left[\{\sin^8\theta - \cos^8\theta\} \frac{1}{\sin... | Mathematics MCQ | Examoogle

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Simplify: $\left[\{\sin^8\theta - \cos^8\theta\} \frac{1}{\sin^4\theta + \cos^4\theta}\right] + 2\cos^2\theta$

Correct Answer

📐 MA • Math Algebraic SimplificationMathematicsTrigonometry

Option D

Quick Summary:

Put $\theta = 0°$ . Then $\sin(0)=0$ and $\cos(0)=1$ . The expression becomes $(0 - 1) \times \frac{1}{0+1} + 2(1) = -1 + 2 = 1$ . Wait, let's re-verify with $\theta = 45°$ .

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📐MAMath SolutionAlgebraic Simplification

📋 Given

An expression containing powers of sine and cosine: [{ $\sin$ ⁸ $\theta -$$\cos$ ⁸ $\theta$ } $\frac{1}{\sin⁴\theta + \cos⁴\theta}$ ] + 2 $\cos$ ² $\theta$

🔢 Formula Used

a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b), a2+b2=(a+b)2−2aba^2 + b^2 = (a+b)^2 - 2aba2+b2=(a+b)2−2ab, sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1sin2θ+cos2θ=1

⚡ Exam Hall Shortcut / Speed Trick

Put $\theta = 0°$ . Then $\sin(0)=0$ and $\cos(0)=1$ . The expression becomes $(0 - 1) \times \frac{1}{0+1} + 2(1) = -1 + 2 = 1$ . Wait, let's re-verify with $\theta = 45°$ .

⚠️ Common Student Trap / Pitfall

Students often fail to expand $\sin^8\theta - \cos^8\theta$ correctly or struggle with the simplification of the denominator $\sin^4\theta + \cos^4\theta$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify the numerator using Difference of Squares

Rewrite $\sin^8\theta - \cos^8\theta$ as $((\sin^4\theta)^2 - (\cos^4\theta)^2)$ , which factors into $(\sin^4\theta - \cos^4\theta)(\sin^4\theta + \cos^4\theta)$

sin⁡8θ−cos⁡8θ=(sin⁡4θ−cos⁡4θ)(sin⁡4θ+cos⁡4θ)\sin^8\theta - \cos^8\theta = (\sin^4\theta - \cos^4\theta)(\sin^4\theta + \cos^4\theta)sin8θ−cos8θ=(sin4θ−cos4θ)(sin4θ+cos4θ)

Cancel the common term

Substitute the factored numerator into the expression. The term $\sin^4\theta + \cos^4\theta$ cancels out, leaving $\sin^4\theta - \cos^4\theta$

(sin⁡4θ−cos⁡4θ)+2cos⁡2θ(\sin^4\theta - \cos^4\theta) + 2\cos^2\theta(sin4θ−cos4θ)+2cos2θ

Further factorize the remaining term

Apply the identity $\sin^4\theta - \cos^4\theta = (\sin^2\theta - \cos^2\theta)(\sin^2\theta + \cos^2\theta)$ . Since $\sin^2\theta + \cos^2\theta = 1$ , this becomes $\sin^2\theta - \cos^2\theta$

(sin⁡2θ−cos⁡2θ)+2cos⁡2θ=sin⁡2θ+cos⁡2θ(\sin^2\theta - \cos^2\theta) + 2\cos^2\theta = \sin^2\theta + \cos^2\theta(sin2θ−cos2θ)+2cos2θ=sin2θ+cos2θ

Calculate the final result

Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ , the result is 1.

111

✅

A is correct because the entire expression simplifies algebraically to 1, following the identity $\sin^2\theta + \cos^2\theta = 1$ .

Core Concepts Used

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Trigonometric Identities Algebraic Factorization Difference of Squares

💡 EXAM TIP

This type of simplification frequently appears in SSC and JEE exams; always test with $\theta = 45°$ if you are stuck to eliminate options quickly.

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