Question

Find the value of the given expression.

$$ 3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x) $$

(A) $1 - 3 \sin^2 x \cos^2 x$ (B) 11 (C) 13 (D) $2 \cos x \sin x - 1$

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Answer


The correct option is (C) 13.

For simplicity, we will replace $x$ with $\theta$. Let's evaluate the given expression:

$$ 3(\sin \theta - \cos \theta)^4 + 6(\sin \theta + \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) $$

We will proceed by breaking down the expression step-by-step.

  1. Expand and simplify $\mathbf{3(\sin \theta - \cos \theta)^4}$:

    $$ 3 \left( \sin \theta - \cos \theta \right)^4 $$

    Using the identity $(\sin \theta - \cos \theta)^4 = (\sin^2 \theta + \cos^2 \theta - 2 \sin \theta \cos \theta)^2$, we have:

    $$ 3 \left( 1 - 2 \sin \theta \cos \theta \right)^2 $$

  2. Expand and simplify $\mathbf{6(\sin \theta + \cos \theta)^2}$:

    $$ 6 \left( \sin \theta + \cos \theta \right)^2 $$

    Using the identity $(\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta$, we have:

    $$ 6 \left( 1 + 2 \sin \theta \cos \theta \right) $$

  3. Expand and simplify $\mathbf{4(\sin^6 \theta + \cos^6 \theta)}$:

    $$ 4 (\sin^6 \theta + \cos^6 \theta) $$

    Using the identity $\sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta + \cos^4 \theta - \sin^2 \theta \cos^2 \theta)$, we get:

    $$ 4 \left( 1 (\sin^4 \theta + \cos^4 \theta + \sin^2 \theta \cos^2 \theta - 3 \sin^2 \theta \cos^2 \theta) \right) $$

  4. Combine all the components:

    $$ 3(1 - 2 \sin \theta \cos \theta)^2 + 6(1 + 2 \sin \theta \cos \theta) + 4( (\sin^4 \theta + \cos^4 \theta - \sin^2 \theta \cos^2 \theta)) $$

    Let's simplify each part:

    $$ = 3(1 - 4 \sin^2 \theta \cos^2 \theta + 4 \sin \theta \cos \theta) $$

    $$

    • 6 + 12 \sin \theta \cos \theta $$

    $$

    • 4 (1 - 3 \sin^2 \theta \cos^2 \theta) $$
  5. Combine and simplify further:

    $$ = 3 (1 - 4 \sin^2 \theta \cos^2 \theta + 4 \sin \theta \cos \theta) $$

    $$

    • 6 (1 + 2 \sin \theta \cos \theta) $$

    $$

    • 4 (1 - 3 \sin^2 \theta \cos^2 \theta) $$

    $$ = 3 + 12 \sin^2 \theta \cos^2 \theta - 12 \sin \theta \cos \theta $$

    $$

    • 6 + 12 \sin \theta \cos \theta $$

    $$

    • 4 - 12 \sin^2 \theta \cos^2 \theta $$
  6. Simplify and finalize:

    $$ = 3 + 12 \sin^2 \theta \cos^2 \theta - 12 \sin \theta \cos \theta $$

    $$

    • 6 + 12 \sin \theta \cos \theta $$

    $$

    • 4 - 12 \sin^2 \theta \cos^2 \theta $$

    Now, combining like terms, we get:

    $$ = 3 + 6 + 4 + 12 \sin^2 \theta \cos^2 \theta - 12 \sin^2 \theta \cos^2 \theta = 13 $$

Therefore, the answer is:

$$ \boxed{13} $$


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