Given that tan x=sqrt3 and cos y=(sqrt2)/(2), 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=sqrt3 and  cos y=(sqrt2)/(2), 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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Find sin x: Use the given information to find the values of sin x and sin y. Since tan x = sqrt(3), we know that sin x / cos x = sqrt(3). We also know that cos^2 x + sin^2 x = 1 (Pythagorean identity). We need to find sin x.Calculate cos x: Calculate cos x using the Pythagorean identity. Since tan x = sin x / cos x = sqrt(3), we can assume a right triangle where the opposite side is sqrt(3) and the adjacent side is 1 (since tan is opposite over adjacent). Therefore, the hypotenuse is sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2. So, cos x = adjacent/hypotenuse = 1/2.Calculate sin x: Calculate sin x using the Pythagorean identity. Now that we have cos x, we can use the identity sin^2 x + cos^2 x = 1 to find sin x. We have cos x = 1/2, so cos^2 x = (1/2)^2 = 1/4. Therefore, sin^2 x = 1 - 1/4 = 3/4, and sin x = sqrt(3/4) = sqrt(3)/2.Find sin y: Use the given information to find sin y. We are given cos y = (sqrt(2))/2. Using the Pythagorean identity sin^2 y + cos^2 y = 1, we can find sin y. We have cos^2 y = ((sqrt(2))/2)^2 = 2/4 = 1/2. Therefore, sin^2 y = 1 - 1/2 = 1/2, and sin y = sqrt(1/2) = sqrt(2)/2.Use angle sum identity: Use the angle sum identity for cosine to find cos(x+y). The angle sum identity for cosine is cos(x+y) = cos x * cos y - sin x * sin y. We have already found cos x, cos y, sin x, and sin y.Substitute and simplify: Substitute the values into the angle sum identity and simplify. cos(x+y) = (1/2) * (sqrt(2)/2) - (sqrt(3)/2) * (sqrt(2)/2) cos(x+y) = (sqrt(2)/4) - (sqrt(6)/4) cos(x+y) = (sqrt(2) - sqrt(6))/4

Given that $\tan x=\sqrt{3}$ and $\cos y=\frac{\sqrt{2}}{2}$ , 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=3 \tan x=\sqrt{3} tanx=3​ and cos⁡y=22 \cos y=\frac{\sqrt{2}}{2} cosy=22​​, and that angles x x x and y y y are both in Quadrant I, find the exact value of cos⁡(x+y) \cos (x+y) cos(x+y), in simplest radical form.\newlineAnswer:

Given that $\tan x=\sqrt{3}$ and $\cos y=\frac{\sqrt{2}}{2}$ , 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: