The given expression is:
[tex]\tan\theta+\cot\theta=\frac{1}{\sin\theta\cos\theta}[/tex]
Rewrite the left side in terms of sin and cosine:
[tex]\begin{gathered} \tan\theta=\frac{\sin\theta}{\cos\theta} \\ \cot\theta=\frac{\cos\theta}{\sin\theta} \\ \tan\theta+\cot\theta=\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta} \end{gathered}[/tex]
Now apply the properties of fractions and solve the addition:
[tex]\begin{gathered} \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=\frac{\sin\theta *\sin\theta+\cos\theta *\cos\theta}{\sin\theta *\cos\theta} \\ =\frac{\sin^2\theta+\cos^2\theta}{\sin\theta *\cos\theta} \end{gathered}[/tex]
Apply the following identity:
[tex]sin^2\theta+cos^2\theta=1[/tex]
Thus:
[tex]\frac{sin^2\theta+cos^2\theta}{sin\theta cos\theta}=\frac{1}{\sin\theta\cos\theta}[/tex]
The verification is o.k.
