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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

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

Convert element to fraction:

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

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

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

1 \cdot (1 + sin (x))

Multiply:

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

= (1 + sin (x))

Remove parentheses:

(a) = a

= 1 + sin (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

= sin (x) (sin (x) + 1)

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

Cancel the common factor:

sin (x) + 1

= sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (4 a) = 2 sin (2 a) cos (2 a)

tan (x) = sec (x) sin (x)

sin (2 D) = 2 cot (D) sin^{2} (D)

tan^{2} (\frac{π}{6}) = \frac{1}{3}

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

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