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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (sin (m) + cos (m)) (sin (m) - cos (m))

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

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

Rewrite using trig identities

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

Expand

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

(sin (m) + cos (m)) (sin (m) - cos (m))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Double Angle identity:

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

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

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

Simplify

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Multiply:

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{\frac{1}{c o s ( x )} - 1}{1 - cos ( x )} = sec (x)

tan (13 5^{\circ} + a) = \frac{tan ( a ) - 1}{tan ( a ) + 1}

tan (\frac{x}{2}) = \frac{tan ( x )}{( sec ( x ) + 1 )}

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

sin (x) - \frac{1}{sin ( x )} = - cos (x) cot (x)

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