Question #2fb5f

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Question #2fb5f

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Answer 1 · 2015-05-15T13:11:05

Apply the Sum and Difference Formulas for Sine and Cosine functions:

$sin (x + y) = sin (x) cos (y) + cos (x) sin (y)$ $sin(x+y) = sin(x)cos(y)+cos(x)sin(y)$
$sin (x - y) = sin (x) cos (y) - cos (x) sin (y)$ $sin(x-y) = sin(x)cos(y)-cos(x)sin(y)$
$cos (x + y) = cos (x) cos (y) - sin (x) sin (y)$ $cos(x+y) =cos(x)cos(y) -sin(x)sin(y)$
$cos (x - y) = cos (x) cos (y) + sin (x) sin (y)$ $cos(x-y) =cos(x)cos(y) +sin(x)sin(y)$

We'll attempt to evaluate the numerator and denominator separately to avoid excessively large expressions:

Numerator:
$sin (x + y) + sin (x - y)$ $color(red)(sin(x+y))+color(blue)(sin(x-y))$

$= sin (x) cos (y) + cos (x) sin (y) + sin (x) cos (y) - cos (x) sin (y)$ $=color(red)(sin(x)cos(y)+cos(x)sin(y))+color(blue)(sin(x)cos(y)-cos(x)sin(y))$

$= 2 sin (x) cos (y)$ $= 2sin(x)cos(y)$

Denominator:
$cos (x + y) + cos (x - y)$ $color(red)(cos(x+y))+color(blue)(cos(x-y))$

$= cos (x) cos (y) - sin (x) sin (y) + cos (x) cos (y) + sin (x) sin (y)$ $=color(red)(cos(x)cos(y) -sin(x)sin(y)) +color(blue)(cos(x)cos(y) +sin(x)sin(y))$

$= 2 cos (x) cos (y)$ $=2cos(x)cos(y)$

Left-side of Equation
$= \frac{Numerator}{Denominator}$ $= "Numerator"/"Denominator"$

$(cancel(2)sin(x)cancel(cos(y)))/(cancel(2)cos(x)cancel(cos(y)))$

$=tan(x)$

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