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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Manipulating right side

1 - tan^{2} (x)

Rewrite using trig identities

1 - tan^{2} (x)

Use the Pythagorean identity:

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplify

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

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

Apply exponent rule:

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

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

Convert element to fraction:

cos^{2} (x) = \frac{cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}, sin^{2} (x) = \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

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

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= cos^{4} (x)

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

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

Expand

sin^{2} (x) (1 - sin^{2} (x)) : sin^{2} (x) - sin^{4} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = sin^{2} (x), b = 1, c = sin^{2} (x)

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

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

Simplify

1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x) : sin^{2} (x) - sin^{4} (x)

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

1 \cdot sin^{2} (x) = sin^{2} (x)

1 \cdot sin^{2} (x)

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= sin^{2} (x)

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

sin^{2} (x) sin^{2} (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

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

Add similar elements:

- sin^{2} (x) + sin^{2} (x) = 0

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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