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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (sin (a) + cos (a)) (sin (a) - cos (a))

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

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

Rewrite using trig identities

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

Expand

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

(sin (a) + cos (a)) (sin (a) - cos (a))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Double Angle identity:

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

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

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

Simplify

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

= - (1 - 2 sin^{2} (a)) \cdot 1

Simplify

- (1 - 2 sin^{2} (a)) \cdot 1 : - 1 + 2 sin^{2} (a)

- (1 - 2 sin^{2} (a)) \cdot 1

Multiply:

(1 - 2 sin^{2} (a)) \cdot 1 = (1 - 2 sin^{2} (a))

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

1 - cos (θ) sin (θ) tan (θ) = cos^{2} (θ)

\frac{1 - sin ^{2} ( A )}{1 - cos ^{2} ( A )} = cot^{2} (A)

\frac{sin ( 6 0 ^{\circ} )}{cos ( 6 0 ^{\circ} )} = tan (6 0^{\circ})

cos (x) csc (x) = cos (x)

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