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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

cos^{4} (x) - sin^{4} (x) + 1

Factor

1 - sin^{4} (x) : - (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

1 - sin^{4} (x)

Factor out common term

- 1

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

Factor

sin^{4} (x) - 1 : (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

sin^{4} (x) - 1

Rewrite

sin^{4} (x) - 1

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

sin^{4} (x) - 1

Rewrite

1

1^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (sin (x) + 1) (sin (x) - 1)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

Expand

(sin (x) + 1) (sin (x) - 1) : sin^{2} (x) - 1

(sin (x) + 1) (sin (x) - 1)

Apply Difference of Two Squares Formula:

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

a = sin (x), b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= sin^{2} (x) - 1

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Apply rule

- (- a) = a

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = cos^{2} (x), b = 2, c = cos^{2} (x)

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

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

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

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

Apply exponent rule:

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

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

= cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= cos^{4} (x)

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

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

Add similar elements:

cos^{4} (x) - cos^{4} (x) = 0

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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