How do you verify $(cos(x))(cos(x)) = 1-sin^2(x)$ ?

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How do you verify $(cos(x))(cos(x)) = 1-sin^2(x)$ ?

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Answer 1 · 2015-11-12T09:24:29

We can rewrite the equation as $cos^2(x)=1-sin^2(x)$ , and thus as

$cos^2(x)+sin^2(x)=1$

This is a fundamental equality for the trigonometric function, and its explanation is very simple: if you consider the unit circle, any point on the circumference $(x,y)$ is of the form $(cos(alpha), sin(alpha)$ , for some angle $alpha$ which identifies the point.

Since $cos(alpha)$ is the $x$ coordinate and $sin(alpha)$ is the $y$ coordinate of the point, we have that $cos(alpha)$ and $sin(alpha)$ are the catheti of a right triangle, whose hypotenusa is the radius, which is one. So, we have that

$cos^2(alpha)+sin^2(alpha) = 1^2 = 1$

I hope this image can help, since it shows that the sine and cosine of an angle form a right triangle whose hypotenusa is the radius. www.geocities.ws

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How do you verify $(cos(x))(cos(x)) = 1-sin^2(x)$ ?

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