Question

\(\sin^6 \theta + \cos^6 \theta + 3\sin^2 \theta \cos^2 \theta =\)

• A. 0
• B. 1
• C. 2
• D. -1

Hint:

The identities for \(\sin^4 \theta + \cos^4 \theta\) and \(\sin^6 \theta + \cos^6 \theta\) are very common in competitive exams. It's useful to remember them:

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

• \(\sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta\)
Memorizing these can save you derivation time during an exam.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to simplify the given trigonometric expression. The expression is a known identity.

Step 2: Key Formula or Approach:
We will use the algebraic identity \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) or \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\).
Let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\).
Then the expression \(\sin^6 \theta + \cos^6 \theta\) becomes \(a^3 + b^3\).
We also know the fundamental trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

Step 3: Detailed Explanation:
Let's rewrite the expression using \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\).
The expression is \(a^3 + b^3 + 3ab\). This does not immediately simplify. Let's use the second identity from Step 2.
We know that \(a+b = \sin^2 \theta + \cos^2 \theta = 1\).
Now, let's use the identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\).
Substitute \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\):
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)^3 - 3(\sin^2 \theta)(\cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), this simplifies to:
\[ \sin^6 \theta + \cos^6 \theta = (1)^3 - 3\sin^2 \theta \cos^2 \theta (1) \]
\[ \sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta \]
Now, substitute this result back into the original expression given in the question:
Original Expression = \((\sin^6 \theta + \cos^6 \theta) + 3\sin^2 \theta \cos^2 \theta\)
\[ = (1 - 3\sin^2 \theta \cos^2 \theta) + 3\sin^2 \theta \cos^2 \theta \]
\[ = 1 \]

Step 4: Final Answer:
The value of the expression is 1.