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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (a) (sec (a) - cos (a))

Express with sin, cos

(- cos (a) + sec (a)) cos (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

- cos (a) cos (a) + 1

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

cos (a) cos (a)

Apply exponent rule:

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

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

= cos^{1 + 1} (a)

Add the numbers:

1 + 1 = 2

= cos^{2} (a)

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (a)

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

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

= 1 - cos^{2} (a)

Rewrite using trig identities

1 - cos^{2} (a)

Use the Pythagorean identity:

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

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

= sin^{2} (a)

= sin^{2} (a)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e tan (x) cos (x) csc (x) = 1

p ro v e \frac{csc ^{4} ( x ) - 1}{1 + csc ^{2} ( x )} = cot^{2} (x)

p ro v e 1 + sin (2 θ) = (cos (θ) sin (θ))^{2}

p ro v e cos^{2} (x) = \frac{1}{2} (cos (2 x) + 1)

p ro v e (csc (x) - 1) (csc (x) + 1) = cot^{2} (x)

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