Find the Exact Value (cos(arcsec(3)+arctan(2)))/(cos(arctan(2)))

$\frac{\cos (arcsec (3) + \arctan (2))}{\cos (\arctan (2))}$

Rewrite $\frac{\cos (arcsec (3) + \arctan (2))}{\cos (\arctan (2))}$ as a product.

$\cos (arcsec (3) + \arctan (2)) \frac{1}{\cos (\arctan (2))}$

Write $\cos (arcsec (3) + \arctan (2))$ as a fraction with denominator $1$ .

$\frac{\cos (arcsec (3) + \arctan (2))}{1} \cdot \frac{1}{\cos (\arctan (2))}$

Simplify.

Divide $\cos (arcsec (3) + \arctan (2))$ by $1$ .

$\cos (arcsec (3) + \arctan (2)) \frac{1}{\cos (\arctan (2))}$

Convert from $\frac{1}{\cos (\arctan (2))}$ to $\sec (\arctan (2))$ .

$\cos (arcsec (3) + \arctan (2)) \sec (\arctan (2))$

Simplify each term.

Evaluate $arcsec (3)$ .

$\cos (1.23095941 + \arctan (2)) \sec (\arctan (2))$

Evaluate $\arctan (2)$ .

$\cos (1.23095941 + 1.10714871) \sec (\arctan (2))$

Simplify the expression.

Add $1.23095941$ and $1.10714871$ .

$\cos (2.33810813) \sec (\arctan (2))$

Draw a triangle in the plane with vertices $(1, 2)$ , $(1, 0)$ , and the origin. Then $\arctan (2)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 2)$ . Therefore, $\sec (\arctan (2))$ is $\sqrt{1 + 2^{2}}$ .

$\cos (2.33810813) \sqrt{1 + 2^{2}}$

Raise $2$ to the power of $2$ .

$\cos (2.33810813) \sqrt{1 + 4}$

Add $1$ and $4$ .

$\cos (2.33810813) \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\cos (arcsec (3) + \arctan (2))}{\cos (\arctan (2))}$

Decimal Form:

$- 1.55228474 \dots$

Find the Exact Value (cos(arcsec(3)+arctan(2)))/(cos(arctan(2)))

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