Is (cos(2x))/((cos(x)+sin(x)))=cos(x)-sin(x) ?

The answer to whether (cos(2x))/((cos(x)+sin(x)))=cos(x)-sin(x) is True

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prove (cos(2x))/((cos(x)+sin(x)))=cos(x)-sin(x)

prove

\frac{cos ( 2 x )}{( cos ( x ) + sin ( x ) )} = cos (x) - sin (x)

True

Solution steps

\frac{cos ( 2 x )}{cos ( x ) + sin ( x )} = cos (x) - sin (x)

Manipulating left side

\frac{cos ( 2 x )}{cos ( x ) + sin ( x )}

Rewrite using trig identities

\frac{cos ( 2 x )}{cos ( x ) + sin ( x )}

Use the Double Angle identity:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

= \frac{cos ^{2} ( x ) - sin ^{2} ( x )}{cos ( x ) + sin ( x )}

Simplify

\frac{cos ^{2} ( x ) - sin ^{2} ( x )}{cos ( x ) + sin ( x )} : cos (x) - sin (x)

\frac{cos ^{2} ( x ) - sin ^{2} ( x )}{cos ( x ) + sin ( x )}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (x) - sin^{2} (x) = (cos (x) + sin (x)) (cos (x) - sin (x))

= \frac{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{cos ( x ) + sin ( x )}

Cancel the common factor:

cos (x) + sin (x)

= cos (x) - sin (x)

= cos (x) - sin (x)

= cos (x) - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (- \frac{3 π}{4})

cos (- \frac{π}{6})

sec (x) = 2, 0 \leq x \leq 2 π

sin (x) = - \frac{2}{2}

cos (4 π)

Is (cos(2x))/((cos(x)+sin(x)))=cos(x)-sin(x) ?
The answer to whether (cos(2x))/((cos(x)+sin(x)))=cos(x)-sin(x) is True