Áp dụng CT cộng:
$\cos a\cos b+\sin a\sin b=\cos(a-b)$
Do đó ta có:
$\cos x\cos\dfrac{x}{2}+\sin x\sin\dfrac{x}{2}= \cos\left(x-\dfrac{x}{2}\right)=\cos\dfrac{x}{2}$
$\cot x+\sin x\left( \dfrac{\cos\dfrac{x}{2}}{\cos x\cos\dfrac{x}{2}}\right)$
$=\cot x+\sin x.\dfrac{1}{\cos x}$
$=\dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}$