Cos(x)=sin(x). Then the value of cos^4x + sin^4x=?

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Cos(x)=sin(x). Then the value of cos^4x + sin^4x=?

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Answer 1 · 2018-05-14T03:05:57

$\frac{1}{2}$ $1/2$

Explanation:

$cos (x) = sin (x)$ $Cos(x)=sin(x)$

Usually I avoid dividing by $sin x$ $sinx$ or $cos x$ $cosx$ because I lose solutions, but we will try solving the equation both ways to see if we get the same solution and all solutions are included:
Divide by $sin x$ $sinx$ on both sides:
$cot x = 1$ $cotx=1$
$x = \frac{π}{4}, \frac{5 π}{4}$ $x=pi/4, (5pi)/4$
$x = \frac{π}{4} + π n$ $x=pi/4+pin$

Divide by $cos x$ $cosx$ on both sides:
$tan x = 1$ $tanx=1$
$x = \frac{π}{4}, \frac{5 π}{4}$ $x=pi/4, (5pi)/4$
$x = \frac{π}{4} + π n$ $x=pi/4+pin$

Yields the same solution either way so:
${(\pm \frac{\sqrt{2}}{2})}^{4} + {(\pm \frac{\sqrt{2}}{2})}^{4} =$ $(+-sqrt2/2)^4 + (+-sqrt2/2)^4=$

$\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ $1/4+1/4=1/2$

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Cos(x)=sin(x). Then the value of cos^4x + sin^4x=?

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