Is cos^2(x)(1-sec^2(x))=-sin^2(x) ?

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

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prove cos^2(x)(1-sec^2(x))=-sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= cos^{2} (x) - 1

= cos^{2} (x) - 1

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

Rewrite using trig identities

- 1 + cos^{2} (x)

Use the Pythagorean identity:

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

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

= - sin^{2} (x)

= - sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (θ) - cos^{2} (θ) = 1

tan^{2} (θ) csc^{2} (θ) = sec^{2} (θ)

(1 + tan (x)) (1 - cot (x)) = tan (x) - cot (x)

(tan (x) + cot (x))^{2} = sec^{4} (x) cot^{2} (x)

(1 + tan^{2} (s)) (1 - sin^{2} (s)) = 1

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