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

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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

csc (x) - \frac{sin ( x )}{1 + cos ( x )} = cot (x)

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

(tan^{2} (x) + 1) (cos^{2} (x) - 1) = - tan^{2} (x)

sec^{2} (θ) (1 - sin^{2} (θ)) = 1

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

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