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 )}

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

p ro v e sin (x) = \frac{1}{csc ( x )}

p ro v e - tan (t) + \frac{cos ( t )}{1 - sin ( t )} = sec (t)

p ro v e cos (x) = sin (\frac{π}{2} - x)

p ro v e cos (x) = sin (x) cot (x)

p ro v e cos^{2} (\frac{θ}{2}) = \frac{sec ( θ ) + 1}{2 sec ( θ )}

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