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.

Step 2: Key Formula or Approach (Alternate Method):
Let a = sin²θ, b = cos²θ. Then a+b=1. Use identity a³+b³ = (a+b)³ - 3ab(a+b) directly.

Step 3: Detailed Explanation:
Let a = sin²θ, b = cos²θ. So a + b = 1. Expression = sin⁶θ + cos⁶θ + 3 sin²θ cos²θ = a³ + b³ + 3ab. Use a³ + b³ = (a+b)³ - 3ab(a+b): a³ + b³ = (1)³ - 3ab(1) = 1 - 3ab. Now expression = (1 - 3ab) + 3ab = 1. All terms with ab cancel out.

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