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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec (a) + tan (a)

Express with sin, cos

sec (a) + tan (a)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Rewrite using trig identities

Multiply by

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

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

Cancel the common factor:

cos (a)

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos (x - \frac{π}{6}) - sin (x + \frac{π}{3}) = 0

cot^{2} (α) + 1 = \frac{1}{sin ^{2} ( α )}

sin (3 x) = 3 sin (x) cos^{2} (x) - sin^{3} (x)

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

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

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