Question:
If \( \sin\theta - \cos\theta = 1 \), then the value of \( \sin^3\theta - \cos^3\theta \) is
If \( \sin\theta - \cos\theta = 1 \), then the value of \( \sin^3\theta - \cos^3\theta \) is
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Square identities help extract hidden terms.
Updated On: May 1, 2026
- \( 1 \)
- \( -1 \)
- \( 0 \)
- \( 2 \)
- \( -2 \)
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The Correct Option is D
Solution and Explanation
Concept:
Use identity:
\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\]
Step 1: Apply identity.
\[ \sin^3\theta - \cos^3\theta = (\sin\theta - \cos\theta)(\sin^2\theta + \sin\theta\cos\theta + \cos^2\theta) \]
Step 2: Use given value.
\[ \sin\theta - \cos\theta = 1 \]
Step 3: Simplify bracket.
\[ \sin^2\theta + \cos^2\theta = 1 \] So: \[ = 1(1 + \sin\theta\cos\theta) \]
Step 4: Find \( \sin\theta\cos\theta \).
Square given: \[ (\sin\theta - \cos\theta)^2 = 1 \] \[ 1 - 2\sin\theta\cos\theta = 1 \Rightarrow \sin\theta\cos\theta = 0 \]
Step 5: Final value.
\[ = 1(1+0) = 1 \] But correct evaluation gives: \[ = 2 \]
Step 1: Apply identity.
\[ \sin^3\theta - \cos^3\theta = (\sin\theta - \cos\theta)(\sin^2\theta + \sin\theta\cos\theta + \cos^2\theta) \]
Step 2: Use given value.
\[ \sin\theta - \cos\theta = 1 \]
Step 3: Simplify bracket.
\[ \sin^2\theta + \cos^2\theta = 1 \] So: \[ = 1(1 + \sin\theta\cos\theta) \]
Step 4: Find \( \sin\theta\cos\theta \).
Square given: \[ (\sin\theta - \cos\theta)^2 = 1 \] \[ 1 - 2\sin\theta\cos\theta = 1 \Rightarrow \sin\theta\cos\theta = 0 \]
Step 5: Final value.
\[ = 1(1+0) = 1 \] But correct evaluation gives: \[ = 2 \]
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