Math / Trigonometry

QuestionA. -8 B. -2 C. -14 D. -4 E.None Q.6) $\cos ^{2}\left(\tan ^{-1} x\right)=$ A. $x^{2}+1$ B. $\frac{x^{2}}{x^{2}+1}$ C. $\frac{1}{x^{2}+1}$ D. $\frac{1}{x^{2}}$ E.None

Studdy Solution

STEP 1

1. We are dealing with a trigonometric expression involving inverse trigonometric functions.
2. The expression $\cos^2(\tan^{-1} x)$ can be simplified using trigonometric identities.

STEP 2

1. Understand the geometric interpretation of $\tan^{-1} x$ .
2. Use trigonometric identities to express $\cos^2(\tan^{-1} x)$ in terms of $x$ .
3. Simplify the expression to match one of the given options.

STEP 3

Consider $\theta = \tan^{-1} x$ . This implies that $\tan \theta = x$ . We can interpret this as a right triangle where the opposite side is $x$ and the adjacent side is $1$ .

STEP 4

Using the Pythagorean theorem, calculate the hypotenuse of the triangle. The hypotenuse $h$ is given by:
$h = \sqrt{x^2 + 1}$

STEP 5

Now, find $\cos \theta$ using the triangle. The cosine of an angle is the ratio of the adjacent side to the hypotenuse:
$\cos \theta = \frac{1}{\sqrt{x^2 + 1}}$

STEP 6

Square the cosine to find $\cos^2 \theta$ :
$\cos^2 \theta = \left(\frac{1}{\sqrt{x^2 + 1}}\right)^2 = \frac{1}{x^2 + 1}$
The expression $\cos^2(\tan^{-1} x)$ simplifies to:
$\boxed{\frac{1}{x^2 + 1}}$
This corresponds to option C.

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QuestionA. -8 B. -2 C. -14 D. -4 E.None Q.6) $\cos ^{2}\left(\tan ^{-1} x\right)=$ A. $x^{2}+1$ B. $\frac{x^{2}}{x^{2}+1}$ C. $\frac{1}{x^{2}+1}$ D. $\frac{1}{x^{2}}$ E.None

Studdy Solution

STEP 1

1. We are dealing with a trigonometric expression involving inverse trigonometric functions.
2. The expression $\cos^2(\tan^{-1} x)$ can be simplified using trigonometric identities.

STEP 2

1. Understand the geometric interpretation of $\tan^{-1} x$ .
2. Use trigonometric identities to express $\cos^2(\tan^{-1} x)$ in terms of $x$ .
3. Simplify the expression to match one of the given options.

STEP 3

Consider $\theta = \tan^{-1} x$ . This implies that $\tan \theta = x$ . We can interpret this as a right triangle where the opposite side is $x$ and the adjacent side is $1$ .

STEP 4

Using the Pythagorean theorem, calculate the hypotenuse of the triangle. The hypotenuse $h$ is given by:
$h = \sqrt{x^2 + 1}$

STEP 5

Now, find $\cos \theta$ using the triangle. The cosine of an angle is the ratio of the adjacent side to the hypotenuse:
$\cos \theta = \frac{1}{\sqrt{x^2 + 1}}$

STEP 6

Square the cosine to find $\cos^2 \theta$ :
$\cos^2 \theta = \left(\frac{1}{\sqrt{x^2 + 1}}\right)^2 = \frac{1}{x^2 + 1}$
The expression $\cos^2(\tan^{-1} x)$ simplifies to:
$\boxed{\frac{1}{x^2 + 1}}$
This corresponds to option C.

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QuestionA. -8 B. -2 C. -14 D. -4 E.None Q.6) cos⁡2(tan⁡−1x)=\cos ^{2}\left(\tan ^{-1} x\right)=cos2(tan−1x)= A. x2+1x^{2}+1x2+1 B. x2x2+1\frac{x^{2}}{x^{2}+1}x2+1x2​ C. 1x2+1\frac{1}{x^{2}+1}x2+11​ D. 1x2\frac{1}{x^{2}}x21​ E.None

Studdy Solution

STEP 1

1. We are dealing with a trigonometric expression involving inverse trigonometric functions.2. The expression cos⁡2(tan⁡−1x)\cos^2(\tan^{-1} x)cos2(tan−1x) can be simplified using trigonometric identities.

STEP 2

1. Understand the geometric interpretation of tan⁡−1x\tan^{-1} xtan−1x.2. Use trigonometric identities to express cos⁡2(tan⁡−1x)\cos^2(\tan^{-1} x)cos2(tan−1x) in terms of xxx.3. Simplify the expression to match one of the given options.

STEP 3

Consider θ=tan⁡−1x\theta = \tan^{-1} xθ=tan−1x. This implies that tan⁡θ=x\tan \theta = xtanθ=x. We can interpret this as a right triangle where the opposite side is xxx and the adjacent side is 111.

STEP 4

Using the Pythagorean theorem, calculate the hypotenuse of the triangle. The hypotenuse hhh is given by:h=x2+1 h = \sqrt{x^2 + 1} h=x2+1​

STEP 5

Now, find cos⁡θ\cos \thetacosθ using the triangle. The cosine of an angle is the ratio of the adjacent side to the hypotenuse:cos⁡θ=1x2+1 \cos \theta = \frac{1}{\sqrt{x^2 + 1}} cosθ=x2+1​1​

STEP 6

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QuestionA. -8 B. -2 C. -14 D. -4 E.None Q.6) $\cos ^{2}\left(\tan ^{-1} x\right)=$ A. $x^{2}+1$ B. $\frac{x^{2}}{x^{2}+1}$ C. $\frac{1}{x^{2}+1}$ D. $\frac{1}{x^{2}}$ E.None

1. We are dealing with a trigonometric expression involving inverse trigonometric functions.
2. The expression $\cos^2(\tan^{-1} x)$ can be simplified using trigonometric identities.

1. Understand the geometric interpretation of $\tan^{-1} x$ .
2. Use trigonometric identities to express $\cos^2(\tan^{-1} x)$ in terms of $x$ .
3. Simplify the expression to match one of the given options.

Consider $\theta = \tan^{-1} x$ . This implies that $\tan \theta = x$ . We can interpret this as a right triangle where the opposite side is $x$ and the adjacent side is $1$ .

Using the Pythagorean theorem, calculate the hypotenuse of the triangle. The hypotenuse $h$ is given by:
$h = \sqrt{x^2 + 1}$

Now, find $\cos \theta$ using the triangle. The cosine of an angle is the ratio of the adjacent side to the hypotenuse:
$\cos \theta = \frac{1}{\sqrt{x^2 + 1}}$