Is cos(2x)=cos^4(x)-sin^4(x) ?

The answer to whether cos(2x)=cos^4(x)-sin^4(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove cos(2x)=cos^4(x)-sin^4(x)

prove

cos (2 x) = cos^{4} (x) - sin^{4} (x)

True

Solution steps

cos (2 x) = cos^{4} (x) - sin^{4} (x)

Manipulating right side

cos^{4} (x) - sin^{4} (x)

Factor

cos^{4} (x) - sin^{4} (x) : (cos^{2} (x) + sin^{2} (x)) (cos (x) + sin (x)) (cos (x) - sin (x))

cos^{4} (x) - sin^{4} (x)

Rewrite

cos^{4} (x) - sin^{4} (x)

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

cos^{4} (x) - sin^{4} (x)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

= cos^{4} (x) - (sin^{2} (x))^{2}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

cos^{4} (x) = (cos^{2} (x))^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + sin (x)) (cos (x) - sin (x))

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= (cos (x) + sin (x)) (cos (x) - sin (x)) \cdot 1

Simplify

= (cos (x) + sin (x)) (cos (x) - sin (x))

Expand

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

(cos (x) + sin (x)) (cos (x) - sin (x))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

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

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

Use the Double Angle identity:

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

= cos (2 x)

= cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

csc (\frac{π}{2} - u) = sec (u)

- 2 sin (k - l) sin (k + l) = cos (2 k) - cos (2 l)

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

tan^{2} (x) + sec^{2} (x) = cot^{2} (x)

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

Is cos(2x)=cos^4(x)-sin^4(x) ?
The answer to whether cos(2x)=cos^4(x)-sin^4(x) is True