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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (sin (y) + cos (y)) (sin (y) - cos (y))

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

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

Rewrite using trig identities

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

Expand

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

(sin (y) + cos (y)) (sin (y) - cos (y))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Double Angle identity:

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

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

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

Simplify

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Multiply:

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

1 - sin^{2} (- θ) = cos^{2} (θ)

tan (θ) csc^{2} (θ) - tan (θ) = cot (θ)

\frac{1 + sec ( θ )}{csc ( θ )} - tan (θ) = sin (θ)

\frac{1 + cot ( x )}{1 + tan ( x )} = cot (x)

cos (α + β) + cos (α - β) = 2 cos (α) cos (β)

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