Is cos(ω)(sec(ω)-cos(ω))=sin^2(ω) ?

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

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prove cos(ω)(sec(ω)-cos(ω))=sin^2(ω)

prove

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

True

Solution steps

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

Manipulating left side

cos (ω) (sec (ω) - cos (ω))

Express with sin, cos

(- cos (ω) + sec (ω)) cos (ω)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

- cos (ω) + \frac{1}{cos ( ω )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

- cos (ω) cos (ω) + 1

cos (ω) cos (ω) = cos^{2} (ω)

cos (ω) cos (ω)

Apply exponent rule:

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

cos (ω) cos (ω) = cos^{1 + 1} (ω)

= cos^{1 + 1} (ω)

Add the numbers:

1 + 1 = 2

= cos^{2} (ω)

= - cos^{2} (ω) + 1

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (ω)

= - - cos^{2} (ω) + 1

= - cos^{2} (ω) + 1

= 1 - cos^{2} (ω)

Rewrite using trig identities

1 - cos^{2} (ω)

Use the Pythagorean identity:

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

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

= sin^{2} (ω)

= sin^{2} (ω)

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( θ )}{1 + cos ( θ )} + cot (θ) = csc (θ)

tan (x) csc (x) = tan (x) sin (x) + cos (x)

(cos (\frac{5 \cdot \frac{π}{3}}{4})) = (cos (5 \frac{π}{12}))

tan (2 1^{\circ}) = 2 tan (t)

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

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