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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec (a) (1 - sin^{2} (a))

Express with sin, cos

(1 - sin^{2} (a)) sec (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

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

Multiply:

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

= (1 - sin^{2} (a))

Remove parentheses:

(a) = a

= 1 - sin^{2} (a)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

cos (a)

= cos (a)

= cos (a)

We showed that the two sides could take the same form

\Rightarrow True

cos (x + y) - cos (x - y) = - 2 sin (x) sin (y)

sec (\frac{π}{2} - u) = csc (u)

\frac{cot ^{3} ( t )}{csc ( t )} = cos (t) (csc^{2} (- 1))

\frac{sec ^{2} ( x ) cot ( x )}{csc ^{2} ( x )} = tan (x)

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

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