Is (cos^2(x)-4)/(cos(x)-2)=cos(x)+2 ?

The answer to whether (cos^2(x)-4)/(cos(x)-2)=cos(x)+2 is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (cos^2(x)-4)/(cos(x)-2)=cos(x)+2

prove

\frac{cos ^{2} ( x ) - 4}{cos ( x ) - 2} = cos (x) + 2

True

Solution steps

\frac{cos ^{2} ( x ) - 4}{cos ( x ) - 2} = cos (x) + 2

Manipulating left side

\frac{cos ^{2} ( x ) - 4}{cos ( x ) - 2}

Simplify

\frac{- 4 + cos ^{2} ( x )}{- 2 + cos ( x )} : cos (x) + 2

\frac{- 4 + cos ^{2} ( x )}{- 2 + cos ( x )}

Factor

- 4 + cos^{2} (x) : (cos (x) + 2) (cos (x) - 2)

- 4 + cos^{2} (x)

Rewrite

4

2^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + 2) (cos (x) - 2)

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

Cancel the common factor:

cos (x) - 2

= cos (x) + 2

= 2 + cos (x)

= cos (x) + 2

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (a) tan^{2} (a) = tan^{2} (a) - sin^{2} (a)

\frac{cos ( a )}{1 - sin ( a )} - tan (a) = sec (a)

tan (θ) (sin (θ) + cot (θ) cos (θ)) = sec (θ)

sin (x + y) = sin (x) + sin (y)

tan (\frac{θ}{2}) + cot (\frac{θ}{2}) = 2 csc (θ)

Is (cos^2(x)-4)/(cos(x)-2)=cos(x)+2 ?
The answer to whether (cos^2(x)-4)/(cos(x)-2)=cos(x)+2 is True