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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

prove

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

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