Given that tan x=6 and cos y=(3)/(sqrt13), and that angles x and y are both in Quadrant I, find the exact value of cos(x+y), in simplest radical form. Answer:

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Given that  tan x=6 and  cos y=(3)/(sqrt13), and that angles  x and  y are both in Quadrant I, find the exact value of  cos(x+y), in simplest radical form. Answer:

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Given Information: We are given tan x = 6 and cos y = 3/√13. Since x and y are in Quadrant I, all trigonometric functions are positive. We need to find cos(x+y). To do this, we will use the cosine addition formula: cos(x+y) = cos x * cos y - sin x * sin y. First, we need to find cos x and sin y.Finding cos x: To find cos x, we use the identity 1 + tan^2 x = sec^2 x. Since tan x = 6, we have 1 + 6^2 = sec^2 x, which simplifies to 1 + 36 = sec^2 x, so sec^2 x = 37. Therefore, sec x = √37, and since sec x is the reciprocal of cos x, we have cos x = 1/√37.Finding sin y: To find sin y, we use the Pythagorean identity sin^2 y + cos^2 y = 1. We know cos y = 3/√13, so cos^2 y = (3/√13)^2 = 9/13. Substituting into the identity, we get sin^2 y + 9/13 = 1. Solving for sin^2 y gives us sin^2 y = 1 - 9/13 = 4/13. Taking the square root, we find sin y = √(4/13) = 2/√13.Substitute into Formula: Now that we have cos x = 1/√37 and sin y = 2/√13, we can substitute these into the cosine addition formula. cos(x+y) = (1/√37) * (3/√13) - (6) * (2/√13). Simplifying this expression, we get cos(x+y) = 3/(√37 * √13) - 12/√13.Combine Terms: To simplify the expression further, we find a common denominator. The common denominator for √37 * √13 and √13 is √37 * √13. So, we rewrite the expression as cos(x+y) = (3√13)/(37 * 13) - (12√37)/(37 * 13).Simplify Denominator: Now we combine the terms over the common denominator: cos(x+y) = (3√13 - 12√37) / (37 * 13). This is the simplified expression for cos(x+y) in radical form.Final Result: Finally, we simplify the denominator: 37 * 13 = 481. So, cos(x+y) = (3√13 - 12√37) / 481. This is the exact value of cos(x+y) in simplest radical form.

Given that $\tan x=6$ and $\cos y=\frac{3}{\sqrt{13}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\cos (x+y)$ , in simplest radical form. $\newline$ Answer:

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Given that $\tan x=6$ and $\cos y=\frac{3}{\sqrt{13}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\cos (x+y)$ , in simplest radical form. $\newline$ Answer: