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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (cos (b) + sin (b)) (cos (b) - sin (b))

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

= (cos (b) + sin (b)) (cos (b) - sin (b))

Expand

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

(cos (b) + sin (b)) (cos (b) - sin (b))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Double Angle identity:

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

= cos (2 b)

= cos (2 b)

We showed that the two sides could take the same form

\Rightarrow True

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