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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

sec (y) + tan (y)

Express with sin, cos

sec (y) + tan (y)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

Multiply by

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

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

Expand

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

(1 + sin (y)) (1 - sin (y))

Apply Difference of Two Squares Formula:

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

a = 1, b = sin (y)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (y)

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

cos (y)

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

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

We showed that the two sides could take the same form

\Rightarrow True

(csc (x) + cot (x)) (sec (x) - 1) = tan (x)

cos (x - \frac{π}{2}) = cos (x) tan (x)

csc (- θ) = - csc (θ)

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

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

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