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题目

解题步骤

1. Understand the question

   simplify the following expression for the given value of x. assume the angle \\(\\theta\\) is acute. \\(\\sqrt{x^2 - 4}\\) if \\(x = -2\\sec \\theta\\)

2. Explanation

   Step1: Substitute $x=-2\sec\theta$

   $$\sqrt{(-2\sec\theta)^2 - 4}$$

   Step2: Simplify the squared term

   $$\sqrt{4\sec^2\theta - 4}$$

   Step3: Factor out the common term

   $$\sqrt{4(\sec^2\theta - 1)}$$

   Step4: Use trigonometric identity $\sec^2\theta-1=\tan^2\theta$

   $$\sqrt{4\tan^2\theta}$$

   Step5: Simplify the square root

   $$2|\tan\theta|$$

   Step6: Use acute $\theta$ property ($\tan\theta>0$)

   $$2\tan\theta$$

3. Final answer

   $2\tan\theta$

答案

Question Analysis

OCR Text

simplify the following expression for the given value of x. assume the angle \\(\\theta\\) is acute. \\(\\sqrt{x^2 - 4}\\) if \\(x = -2\\sec \\theta\\)

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