Question

simplify the expression.
±√(1 - cos 32θ)/(1 + cos 32θ)
±√(1 - cos 32θ)/(1 + cos 32θ)=□
(simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the trigonometric identity

We know the identity \(\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{1+\cos\alpha}}\). Here \(\alpha = 32\theta\).

Step2: Substitute \(\alpha\) into the identity

Substituting \(\alpha = 32\theta\) into the identity \(\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{1+\cos\alpha}}\), we get \(\pm\sqrt{\frac{1-\cos32\theta}{1 + \cos32\theta}}=\tan\frac{32\theta}{2}\).

Step3: Simplify the expression

Simplify \(\frac{32\theta}{2}\), we have \(\tan16\theta\).

Answer:

\(\tan16\theta\)