Question:
If
\[
\sin\theta+\cos\theta=\sqrt2
\]
then
\[
\sin\theta\cos\theta=
\]
If
\[
\sin\theta+\cos\theta=\sqrt2
\]
then
\[
\sin\theta\cos\theta=
\]
Show Hint
Whenever \((\sin\theta+\cos\theta)\) is given, squaring helps apply the identity \(\sin^2\theta+\cos^2\theta=1\).
Updated On: Jul 15, 2026
- \(2\)
- \(\frac12\)
- \(2\sqrt2\)
- \(\frac13\)
Show Solution
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The Correct Option is B
Solution and Explanation
Square both sides:
\[
(\sin\theta+\cos\theta)^2=2
\]
Expand:
\[
\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=2
\]
Using identity:
\[
\sin^2\theta+\cos^2\theta=1
\]
So:
\[
1+2\sin\theta\cos\theta=2
\]
\[
2\sin\theta\cos\theta=1
\]
\[
\sin\theta\cos\theta=\frac12
\]
Thus,
\[
\boxed{\frac12}
\]
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