A Beginner's Guide To Understanding cos(arctan(x))

How do you simplify cos(arctan(x))?

The term "cos(arctan)" refers to the cosine of the inverse of the tangent function. The inverse tangent function, also known as the arctangent, is the function that "undoes" the tangent function. In other words, if you apply the tangent function to a number and then apply the arctangent to the result, you will get the original number back. The cosine function is one of the basic trigonometric functions, and it is defined as the ratio of the side adjacent to an angle in a right triangle to the hypotenuse of the triangle. When you apply the cosine function to the result of the arctangent function, you are finding the cosine of the angle that would produce the original number when the tangent function is applied to it. This can be useful in a variety of mathematical contexts.

Let y = arctan(x)

Therefore x = tan(y)

Since we let y=arctan(x) our problem goes from finding cos(arctan(x)) to finding cos(y).

So if x=tan(y) we need to find cos(y).

tan = opposite/adjacent in a right angled triangle.

So we can re-write tan(y) explicitly as tan(y)=x/1 and we can see that x=opposite and 1=adjacent in a right angled triangle.

Hence, using Pythagoras, we can find the missing side (the hypotenuse).

Hypotenuse = root{x²+1}

Now we can find cos(y).

cos(y)=adjacent/hypotenuse.

Therefore cos(y)=1/rt{x²+1}

Finally we can conclude that cos(arctan(x))=1/rt{x ²+1}

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