Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]

Guida verificata passo dopo passo

Recognize that the expression involves the square root of a fraction with cosine terms: \(\sqrt{\frac{1 + \cos 165^\circ}{1 - \cos 165^\circ}}\).

Recall the trigonometric identity for tangent in terms of cosine: \(\tan^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{1 + \cos \theta}\), which can be rearranged to relate to the given expression.

Rewrite the expression inside the square root by comparing it to the identity: \(\frac{1 + \cos 165^\circ}{1 - \cos 165^\circ} = \frac{1}{\frac{1 - \cos 165^\circ}{1 + \cos 165^\circ}}\) and recognize this as the reciprocal of \(\tan^2 \frac{165^\circ}{2}\).

Simplify the square root of the reciprocal of \(\tan^2 \frac{165^\circ}{2}\) to get \(\frac{1}{\tan \frac{165^\circ}{2}}\), which is equal to \(\cot \frac{165^\circ}{2}\).

Evaluate or express the final simplified form as \(\cot 82.5^\circ\), which is the simplified trigonometric expression.

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