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Question Find the secant squared of the inverse cotangent of xx. Select the correct answer.

Studdy Solution

STEP 1

Assumptions
1. We are looking to find the value of sec⁡2(cot⁡−1x)\sec^2(\cot^{-1} x).
2. cot⁡−1x\cot^{-1} x represents the inverse cotangent function, which gives us an angle whose cotangent is xx.
3. The relationship between the trigonometric functions cotangent and secant will be used.
4. We will use the identity sec⁡2θ=1+tan⁡2θ\sec^2 \theta = 1 + \tan^2 \theta.

STEP 2

First, let's denote the angle whose cotangent is xx as θ\theta. Therefore, we have:
cot⁡θ=x\cot \theta = x

STEP 3

We know that the cotangent of an angle in a right triangle is the adjacent side over the opposite side. Let's represent the adjacent side as xx and the opposite side as 11 (since cot⁡θ=adjacentopposite=x\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = x).

STEP 4

Using the Pythagorean theorem, we can find the hypotenuse hh of the right triangle with sides xx and 11.
h2=x2+12h^2 = x^2 + 1^2

STEP 5

Solve for the hypotenuse hh.
h=x2+1h = \sqrt{x^2 + 1}

STEP 6

Now, we can express the secant of θ\theta in terms of xx and hh. Secant is the reciprocal of cosine, and cosine is the adjacent side over the hypotenuse. Therefore:
sec⁡θ=1cos⁡θ=hx\sec \theta = \frac{1}{\cos \theta} = \frac{h}{x}

STEP 7

Substitute the value of hh into the expression for sec⁡θ\sec \theta.
sec⁡θ=x2+1x\sec \theta = \frac{\sqrt{x^2 + 1}}{x}

STEP 8

Square the expression to find sec⁡2θ\sec^2 \theta.
sec⁡2θ=(x2+1x)2\sec^2 \theta = \left(\frac{\sqrt{x^2 + 1}}{x}\right)^2

STEP 9

Simplify the squared expression.
sec⁡2θ=x2+1x2\sec^2 \theta = \frac{x^2 + 1}{x^2}

STEP 10

Now, we replace θ\theta with cot⁡−1x\cot^{-1} x to find sec⁡2(cot⁡−1x)\sec^2(\cot^{-1} x).
sec⁡2(cot⁡−1x)=x2+1x2\sec^2(\cot^{-1} x) = \frac{x^2 + 1}{x^2}

STEP 11

We can see that the expression matches one of the given options.
The solution is:
sec⁡2(cot⁡−1x)=x2+1x2\sec^2(\cot^{-1} x) = \frac{x^2 + 1}{x^2}
Therefore, the correct answer is 1+x2x2\frac{1+x^2}{x^2}.

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