Is 1-((sin^2(x)))/((1+cos(x)))=cos(x) ?

The answer to whether 1-((sin^2(x)))/((1+cos(x)))=cos(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove 1-((sin^2(x)))/((1+cos(x)))=cos(x)

prove

1 - \frac{( sin ^{2} ( x ) )}{( 1 + cos ( x ) )} = cos (x)

True

Solution steps

1 - \frac{sin ^{2} ( x )}{1 + cos ( x )} = cos (x)

Manipulating left side

1 - \frac{sin ^{2} ( x )}{1 + cos ( x )}

Simplify

1 - \frac{sin ^{2} ( x )}{1 + cos ( x )} : \frac{1 + cos ( x ) - sin ^{2} ( x )}{1 + cos ( x )}

1 - \frac{sin ^{2} ( x )}{1 + cos ( x )}

Convert element to fraction:

1 = \frac{1 ( 1 + cos ( x ) )}{1 + cos ( x )}

= \frac{1 \cdot ( 1 + cos ( x ) )}{1 + cos ( x )} - \frac{sin ^{2} ( x )}{1 + cos ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot ( 1 + cos ( x ) ) - sin ^{2} ( x )}{1 + cos ( x )}

1 \cdot (1 + cos (x)) = 1 + cos (x)

1 \cdot (1 + cos (x))

Multiply:

1 \cdot (1 + cos (x)) = (1 + cos (x))

= (1 + cos (x))

Remove parentheses:

(a) = a

= 1 + cos (x)

= \frac{1 + cos ( x ) - sin ^{2} ( x )}{1 + cos ( x )}

= \frac{1 + cos ( x ) - sin ^{2} ( x )}{1 + cos ( x )}

Rewrite using trig identities

\frac{1 + cos ( x ) - sin ^{2} ( x )}{1 + cos ( x )}

Use the Pythagorean identity:

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

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

= \frac{cos ( x ) + cos ^{2} ( x )}{1 + cos ( x )}

Simplify

\frac{cos ( x ) + cos ^{2} ( x )}{1 + cos ( x )} : cos (x)

\frac{cos ( x ) + cos ^{2} ( x )}{1 + cos ( x )}

Factor

cos (x) + cos^{2} (x) : cos (x) (1 + cos (x))

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

= cos (x) + cos (x) cos (x)

Factor out common term

cos (x)

= cos (x) (1 + cos (x))

= \frac{cos ( x ) ( 1 + cos ( x ) )}{1 + cos ( x )}

Cancel the common factor:

1 + cos (x)

= cos (x)

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (2 x) = 2 \cdot sin (x) \cdot cos (x)

3 tan (A) = \frac{1}{tan ( B )}

\frac{1}{sec ^{2} ( x )} = sin^{2} (x) + cos (2 x)

csc (2 y) = csc (y) 7

csc^{2} (θ) (1 - cos^{2} (θ)) = cos (\frac{π}{2})

Is 1-((sin^2(x)))/((1+cos(x)))=cos(x) ?
The answer to whether 1-((sin^2(x)))/((1+cos(x)))=cos(x) is True