To verify the given trigonometric identity [tex]\(\frac{1}{\tan \theta + \cot \theta} = \sin \theta \cdot \cos \theta\)[/tex], let us work through it step-by-step.
1. Express [tex]\(\tan \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] in terms of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
2. Substitute the expressions for [tex]\(\tan \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] into the original equation:
[tex]\[
\frac{1}{\tan \theta + \cot \theta} = \frac{1}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}
\][/tex]
3. Combine the fractions in the denominator:
[tex]\[
\frac{1}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}} = \frac{1}{\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}}
\][/tex]
4. Recognize the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[
\frac{1}{\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}} = \frac{1}{\frac{1}{\sin \theta \cos \theta}}
\][/tex]
5. Simplify the expression:
[tex]\[
\frac{1}{\frac{1}{\sin \theta \cos \theta}} = \sin \theta \cos \theta
\][/tex]
Therefore, we have shown that both sides of the given expression are equal:
[tex]\[
\frac{1}{\tan \theta + \cot \theta} = \sin \theta \cdot \cos \theta
\][/tex]
Thus, the equation is verified:
[tex]\[
\frac{1}{\tan \theta + \cot \theta} = \sin \theta \cdot \cos \theta
\][/tex]
