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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Cancel

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

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

Factor

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

1 - sin^{2} (u)

Factor out common term

- 1

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

Factor

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

sin^{2} (u) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (sin (u) + 1) (sin (u) - 1)

= - (sin (u) + 1) (sin (u) - 1)

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

Cancel

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

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

- sin (u) + 1 = - (sin (u) - 1)

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

Refine

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

Cancel the common factor:

sin (u) - 1

= sin (u) + 1

= sin (u) + 1

= 1 - (sin (u) + 1)

- (sin (u) + 1) : - sin (u) - 1

- (sin (u) + 1)

Distribute parentheses

= - (sin (u)) - (1)

Apply minus-plus rules

+ (- a) = - a

= - sin (u) - 1

= 1 - sin (u) - 1

Simplify

1 - sin (u) - 1 : - sin (u)

1 - sin (u) - 1

Group like terms

= - sin (u) + 1 - 1

1 - 1 = 0

= - sin (u)

= - sin (u)

= - sin (u)

= - sin (u)

We showed that the two sides could take the same form

\Rightarrow True

tan (θ) (cot (θ) cos (θ) + sin (θ)) = sec (θ)

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

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

sin (y) sec (y) = tan (y)

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

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