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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

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

Convert element to fraction:

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

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

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 ( a ) ) - cos ^{2} ( a )}{1 + sin ( a )}

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

1 \cdot (1 + sin (a))

Multiply:

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

= (1 + sin (a))

Remove parentheses:

(a) = a

= 1 + sin (a)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Factor

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

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

Apply exponent rule:

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

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

= sin (a) sin (a) + sin (a)

Factor out common term

sin (a)

= sin (a) (sin (a) + 1)

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

Cancel the common factor:

sin (a) + 1

= sin (a)

= sin (a)

= sin (a)

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( x ) csc ( x )}{sec ( x )} = 1

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

2 tan (θ) (1 + cos (2 θ)) = 2 sin (2 θ)

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

tan (a) + cot (a) = 2 csc (2 a)

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