If \(\tan(x)+\cot(x)=2\), find \(\sin(x)\cos(x)\).
1/2
We are given \(\tan x+\cot x=2\) and asked to find the product \(\sin x\cos x\).
Convert both ratios to sine and cosine: \(\tan x=\frac{\sin x}{\cos x}\) and \(\cot x=\frac{\cos x}{\sin x}\).
Add them over the common denominator \(\sin x\cos x\): \(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{\sin^{2}x+\cos^{2}x}{\sin x\cos x}\).
Use the Pythagorean identity \(\sin^{2}x+\cos^{2}x=1\), so the whole left side collapses to \(\frac{1}{\sin x\cos x}\).
Set this equal to the given value: \(\frac{1}{\sin x\cos x}=2\).
Take reciprocals of both sides to isolate the product: \(\sin x\cos x=\frac{1}{2}\).
The value of \(\sin x\cos x\) is 1/2.
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