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find the vertical asymptotes of the function y=sqrt((1-sin x)/(1+sin x))

Find the vertical asymptotes of the functiony=1−sin⁡x1+sin⁡xy=\sqrt{\frac{1-\sin x}{1+\sin x}}

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Full solution

Q. Find the vertical asymptotes of the functiony=1−sin⁡x1+sin⁡xy=\sqrt{\frac{1-\sin x}{1+\sin x}}
  1. Simplify using trigonometric identity: Simplify the expression inside the square root using the trigonometric identity for tangent half-angle.\newlineCalculation: \newlineUsing the identity, (1−sin⁡x)/(1+sin⁡x)=(tan⁡(x/2))2(1 - \sin x) / (1 + \sin x) = (\tan(x/2))^2.
  2. Substitute simplified expression: Substitute the simplified expression back into the original equation.\newlineCalculation:\newliney=(tan⁡(x2))2y = \sqrt{(\tan(\frac{x}{2}))^2}.
  3. Simplify square root: Simplify the square root of a square.\newlineCalculation:\newliney=∣tan⁡(x2)∣y = |\tan(\frac{x}{2})|.

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