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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

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

Convert element to fraction:

1 = \frac{1 ( 1 + cos ( t ) )}{1 + cos ( t )}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

1 \cdot (1 + cos (t)) = 1 + cos (t)

1 \cdot (1 + cos (t))

Multiply:

1 \cdot (1 + cos (t)) = (1 + cos (t))

= (1 + cos (t))

Remove parentheses:

(a) = a

= 1 + cos (t)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

\frac{cos ( t ) + cos ^{2} ( t )}{1 + cos ( t )} : cos (t)

\frac{cos ( t ) + cos ^{2} ( t )}{1 + cos ( t )}

Factor

cos (t) + cos^{2} (t) : cos (t) (1 + cos (t))

cos (t) + cos^{2} (t)

Apply exponent rule:

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

cos^{2} (t) = cos (t) cos (t)

= cos (t) + cos (t) cos (t)

Factor out common term

cos (t)

= cos (t) (1 + cos (t))

= \frac{cos ( t ) ( 1 + cos ( t ) )}{1 + cos ( t )}

Cancel the common factor:

1 + cos (t)

= cos (t)

= cos (t)

= cos (t)

We showed that the two sides could take the same form

\Rightarrow True

(csc (α) + cot (α)) (sec (α) - 1) = tan (α)

1 - 2 sin^{2} (θ) = 2 cos^{2} (θ) - 1

sin (\frac{π}{4}) = cos (\frac{π}{4})

sec^{2} (x) cot^{2} (x) - 1 = cot^{2} (x)

sin (3 a) + sin (a) = 4 sin (a) - 4 sin (3 a)

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