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

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

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

prove

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

True

Solution steps

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

Manipulating right side

cos (x) (sec (x) - cos (x))

Express with sin, cos

(- cos (x) + sec (x)) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

- cos (x) cos (x) + 1

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (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

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

(1 - tan^{2} (x)) = sec^{2} (x) - 2 tan (x)

(\frac{1}{2} sin (2 x)) (1 - cos (4 x)) = sin^{3} (2 x)

81 - 9 (3 sin (x))^{2} = 11

sin^{3} (2 x) = \frac{1}{2} sin (2 x) (1 - cos (4 x))

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