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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin^{2} (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = - cos^{2} (x), c = 1, d = - cos^{2} (x)

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

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

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

Simplify

- 1 \cdot 1 + 1 \cdot cos^{2} (x) - 1 \cdot cos^{2} (x) + cos^{2} (x) cos^{2} (x) : - 1 + cos^{4} (x)

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

Add similar elements:

1 \cdot cos^{2} (x) - 1 \cdot cos^{2} (x) = 0

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

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

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)

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

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

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

= cos^{4} (x) - 1

We showed that the two sides could take the same form

\Rightarrow True

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p ro v e \frac{cot ( x )}{csc ( x ) - sin ( x )} = sec (x)

p ro v e sin (x) = 5 cos (x)

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