The value of cos(tan^(-1)(x)) is:

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Question: The value of cos(tan^(-1)(x)) is:

Options:

Correct Answer: 1/√(1+x^2)

Solution:

Using the right triangle definition, cos(tan^(-1)(x)) = adjacent/hypotenuse = 1/√(1+x^2).

The value of cos(tan^(-1)(x)) is:

The value of cos(tan^(-1)(x)) is:

Step 1: Understand that tan^(-1)(x) is the angle whose tangent is x.
Step 2: Draw a right triangle where the opposite side is x and the adjacent side is 1. This is because tangent is defined as opposite/adjacent.
Step 3: Use the Pythagorean theorem to find the hypotenuse. The hypotenuse is √(1^2 + x^2) = √(1 + x^2).
Step 4: Recall that cosine is defined as adjacent/hypotenuse. In our triangle, the adjacent side is 1 and the hypotenuse is √(1 + x^2).
Step 5: Write the expression for cosine: cos(tan^(-1)(x)) = adjacent/hypotenuse = 1/√(1 + x^2).

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