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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Cancel

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

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

Factor

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

1 - sin^{2} (x)

Factor out common term

- 1

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

Factor

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

sin^{2} (x) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

Cancel

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

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

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

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

Refine

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

Cancel the common factor:

sin (x) - 1

= sin (x) + 1

= sin (x) + 1

= 1 - (sin (x) + 1)

- (sin (x) + 1) : - sin (x) - 1

- (sin (x) + 1)

Distribute parentheses

= - (sin (x)) - (1)

Apply minus-plus rules

+ (- a) = - a

= - sin (x) - 1

= 1 - sin (x) - 1

Simplify

1 - sin (x) - 1 : - sin (x)

1 - sin (x) - 1

Group like terms

= - sin (x) + 1 - 1

1 - 1 = 0

= - sin (x)

= - sin (x)

= - sin (x)

= - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{4} (a) - cos^{4} (a) = 2 sin^{2} (a) - 1

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

1 - cos (θ) sin (θ) tan (θ) = cos^{2} (θ)

\frac{1 - sin ^{2} ( A )}{1 - cos ^{2} ( A )} = cot^{2} (A)

\frac{sin ( 6 0 ^{\circ} )}{cos ( 6 0 ^{\circ} )} = tan (6 0^{\circ})

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