Is (sec(a)+tan(a))(1-sin(a))=cos(a) ?

The answer to whether (sec(a)+tan(a))(1-sin(a))=cos(a) is True

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prove (sec(a)+tan(a))(1-sin(a))=cos(a)

prove

(sec (a) + tan (a)) (1 - sin (a)) = cos (a)

True

Solution steps

(sec (a) + tan (a)) (1 - sin (a)) = cos (a)

Manipulating left side

(sec (a) + tan (a)) (1 - sin (a))

Express with sin, cos

(1 - sin (a)) (sec (a) + tan (a))

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Simplify

\frac{1}{cos ( a )} + \frac{sin ( a )}{cos ( a )} : \frac{sin ( a ) + 1}{cos ( a )}

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

Apply rule

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

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

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

Multiply fractions:

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

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

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

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

Rewrite using trig identities

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

Expand

(1 + sin (a)) (1 - sin (a)) : 1 - sin^{2} (a)

(1 + sin (a)) (1 - sin (a))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (a)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (a)

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

Use the Pythagorean identity:

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

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

= \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

tan^{2} (a) + 1 = sec^{2} (a)

\frac{8 csc ( - x )}{sec ( - x )} = - 8 cot (x)

cos (α - β) = cos (α) cos (β) + sin (α) sin (β)

\frac{4 sin ^{2} ( t )}{tan ^{2} ( t )} = 4 cos^{2} (t)

\frac{csc ^{2} ( t )}{csc ^{2} ( t ) - 2} = sec (2 t)

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