Question:
If
\[
\sin\theta+\cos\theta=1,
\]
then the value of
\[
\sin\theta\cos\theta
\]
is:
If
\[
\sin\theta+\cos\theta=1,
\]
then the value of
\[
\sin\theta\cos\theta
\]
is:
Show Hint
Whenever an expression contains \(\sin\theta+\cos\theta\), try squaring it immediately. The identity
\[
\sin^2\theta+\cos^2\theta=1
\]
usually simplifies the problem dramatically.
Updated On: Jun 10, 2026
- \(-1\)
- \(0\)
- \(\frac12\)
- \(1\)
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
Expressions involving \(\sin\theta+\cos\theta\) are often simplified by squaring both sides. This is because the identity
\[
\sin^2\theta+\cos^2\theta=1
\]
appears naturally after squaring and helps in determining the product \(\sin\theta\cos\theta\).
This technique is one of the most frequently used methods in trigonometric simplification problems.
Step 1: Write the given relation. We are given \[ \sin\theta+\cos\theta=1. \]
Step 2: Square both sides. Squaring, \[ (\sin\theta+\cos\theta)^2=1^2. \] \[ \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1. \]
Step 3: Apply the fundamental identity. Since \[ \sin^2\theta+\cos^2\theta=1, \] the equation becomes \[ 1+2\sin\theta\cos\theta=1. \]
Step 4: Solve for the product. Subtracting \(1\) from both sides, \[ 2\sin\theta\cos\theta=0. \] Therefore, \[ \sin\theta\cos\theta=0. \]
Step 5: Verification. The result satisfies the transformed equation exactly. Hence there is no contradiction and the answer is valid.
Step 6: Final Conclusion. \[ \boxed{\sin\theta\cos\theta=0} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
Step 1: Write the given relation. We are given \[ \sin\theta+\cos\theta=1. \]
Step 2: Square both sides. Squaring, \[ (\sin\theta+\cos\theta)^2=1^2. \] \[ \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1. \]
Step 3: Apply the fundamental identity. Since \[ \sin^2\theta+\cos^2\theta=1, \] the equation becomes \[ 1+2\sin\theta\cos\theta=1. \]
Step 4: Solve for the product. Subtracting \(1\) from both sides, \[ 2\sin\theta\cos\theta=0. \] Therefore, \[ \sin\theta\cos\theta=0. \]
Step 5: Verification. The result satisfies the transformed equation exactly. Hence there is no contradiction and the answer is valid.
Step 6: Final Conclusion. \[ \boxed{\sin\theta\cos\theta=0} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
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