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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec (u) - tan (u)

Express with sin, cos

sec (u) - tan (u)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( u )} - tan (u)

Use the basic trigonometric identity:

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

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

Simplify

\frac{1}{cos ( u )} - \frac{sin ( u )}{cos ( u )} : \frac{1 - sin ( u )}{cos ( u )}

\frac{1}{cos ( u )} - \frac{sin ( u )}{cos ( u )}

Apply rule

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

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

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

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

Rewrite using trig identities

Multiply by

\frac{1 + sin ( u )}{1 + sin ( u )}

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

Expand

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

(1 + sin (u)) (1 - sin (u))

Apply Difference of Two Squares Formula:

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

a = 1, b = sin (u)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (u)

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

Use the Pythagorean identity:

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

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

= \frac{cos ^{2} ( u )}{( 1 + sin ( u ) ) cos ( u )}

Cancel the common factor:

cos (u)

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e cot (x) + \frac{sin ( x )}{1 + cos ( x )} = csc (x)

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

p ro v e cot (x) (sec^{2} (x) - 1) = tan (x)

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

p ro v e cot (2 θ) = \frac{cot ^{2} ( θ ) - 1}{2 cot ( θ )}

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