Answer
$\sin\theta(\cot\theta+\tan\theta)=\sec\theta$
Work Step by Step
$\sin\theta(\cot\theta+\tan\theta)=\sec\theta$
Substitute $\cot\theta$ by $\dfrac{\cos\theta}{\sin\theta}$ and $\tan\theta$ by $\dfrac{\sin\theta}{\cos\theta}$:
$\sin\theta\Big(\dfrac{\cos\theta}{\sin\theta}+\dfrac{\sin\theta}{\cos\theta}\Big)=\sec\theta$
Evaluate the sum of fractions inside the parentheses:
$\sin\theta\Big(\dfrac{\sin^{2}\theta+\cos^{2}\theta}{\sin\theta\cos\theta}\Big)=\sec\theta$
Evaluate the product on the left side:
$\dfrac{\sin^{2}\theta+\cos^{2}\theta}{\cos\theta}=\sec\theta$
Substitute $\sin^{2}\theta+\cos^{2}\theta$ by $1$:
$\dfrac{1}{\cos\theta}=\sec\theta$
Since $\dfrac{1}{\cos\theta}=\sec\theta$, the identity is proved
