How do you simplify $sin(arccos(x))$ ?

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Answer 1 · 2016-10-21T22:07:41

$sin(arccos(x)) = sqrt(1-x^2)$

From Pythagoras, we have:

$sin^2 theta + cos^2 theta = 1$

If $x in [-1, 1]$ and $theta = arccos(x)$ then:

$theta in [0, pi]$

$sin(theta) >= 0$

Hence:

$sin(arccos(x)) = sin(theta) = sqrt(1 - cos^2 theta) = sqrt(1-x^2)$

Note we can use the non-negative square root since we have already established that $sin(arccos(x)) >= 0$

How do you simplify sin(arccos(x))sin(arccos(x))?