Is sin^4(x)-cos^4(x)=1-2cos^2(x) ?

The answer to whether sin^4(x)-cos^4(x)=1-2cos^2(x) is True

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prove sin^4(x)-cos^4(x)=1-2cos^2(x)

prove

sin^{4} (x) - cos^{4} (x) = 1 - 2 cos^{2} (x)

True

Solution steps

sin^{4} (x) - cos^{4} (x) = 1 - 2 cos^{2} (x)

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Expand

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

Simplify

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

Simplify

- (2 cos^{2} (x) - 1) \cdot 1 : - 2 cos^{2} (x) + 1

- (2 cos^{2} (x) - 1) \cdot 1

Multiply:

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e sec^{4} (x) - tan^{4} (x) = 1 + 2 tan^{2} (x)

p ro v e sin (- x) = - sin (x)

p ro v e sin (x + π) = - sin (x)

p ro v e \frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

p ro v e tan^{5} (x) = tan^{3} (x) sec^{2} (x) - tan^{3} (x)

Is sin^4(x)-cos^4(x)=1-2cos^2(x) ?
The answer to whether sin^4(x)-cos^4(x)=1-2cos^2(x) is True