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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (cos (t) + sin (t)) (cos (t) - sin (t))

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

= (cos (t) + sin (t)) (cos (t) - sin (t))

Expand

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

(cos (t) + sin (t)) (cos (t) - sin (t))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Double Angle identity:

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

= cos (2 t)

Use the Double Angle identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

p ro v e \frac{tan ( x ) + tan ( y )}{cot ( x ) + cot ( y )} = tan (x) tan (y)

p ro v e \frac{cos ( \frac{π}{2} + x )}{cos ( π + x )} = tan (x)

p ro v e 1 - cos (2 x) = tan (x) sin (2 x)

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

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