How do you prove $cosx=pi/2 - sinx$ ?

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Answer 1 · 2016-03-01T21:31:20

The closest true identities to the given equality would be

$cos(x-pi/2) = sin(x)$
or
$cos(x) = sin(x+pi/2)$

(These are equivalent, as you could, for example, substitute $x = y+pi/2$ into the first to obtain the second)

To prove the above, we can use the identity

$cos(alpha + beta) = cos(alpha)cos(beta)-sin(alpha)sin(beta)$

along with sine being an odd function and cosine being an even function. That is
$sin(-x) = -sin(x)$
and
$cos(-x) = cos(x)$

With these, we have

$cos(x-pi/2) = cos(x)cos(-pi/2)-sin(x)sin(-pi/2)$

$=cos(x)cos(pi/2)+sin(x)sin(pi/2)$

$=cos(x)*0+sin(x)*1$

$=sin(x)$

How do you prove cosx=pi/2 - sinx cosx=pi/2 - sinx ?