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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

\frac{1 - cos ^{4} ( x )}{1 - cos ^{2} ( x )}

Factor

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

1 - cos^{4} (x)

Factor out common term

- 1

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

Factor

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

cos^{4} (x) - 1

Rewrite

cos^{4} (x) - 1

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

cos^{4} (x) - 1

Rewrite

1

1^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

cos^{2} (x) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + 1) (cos (x) - 1)

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

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

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

Factor

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

1 - cos^{2} (x)

Factor out common term

- 1

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

Factor

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

cos^{2} (x) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + 1) (cos (x) - 1)

= - (cos (x) + 1) (cos (x) - 1)

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

Cancel

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

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

Cancel the common factor:

cos (x) + 1

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

Cancel the common factor:

cos (x) - 1

= cos^{2} (x) + 1

= cos^{2} (x) + 1

= cos^{2} (x) + 1

= cos^{2} (x) + 1

= 1 + cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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