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

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

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

1 - cos (x)

Multiply by

\frac{1 + cos ( x )}{1 + cos ( x )}

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

Apply Difference of Two Squares Formula:

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

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

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

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

(1 - cos (x)) (1 + sec (x)) = - 1

2 cos^{2} (a) = (1 + sec (2 a)) cos (2 a)

sec (a) - sin (a) tan (a) = cos (a)

\frac{cos ( t )}{sec ( t ) - tan ( t )} = 1 + sin (t)

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