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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}} - \frac{sin ( u )}{1 + cot ( u )}

Use the basic trigonometric identity:

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

= \frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}} - \frac{sin ( u )}{1 + \frac{c o s ( u )}{s i n ( u )}}

Simplify

\frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}} - \frac{sin ( u )}{1 + \frac{c o s ( u )}{s i n ( u )}} : cos (u) - sin (u)

\frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}} - \frac{sin ( u )}{1 + \frac{c o s ( u )}{s i n ( u )}}

\frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}} = \frac{cos ^{2} ( u )}{cos ( u ) + sin ( u )}

\frac{cos ( u )}{1 + \frac{s i n ( u )}{c o s ( u )}}

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (u) = cos (u)

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

= \frac{cos ( u )}{\frac{c o s ( u ) + s i n ( u )}{c o s ( u )}}

Apply the fraction rule:

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

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

cos (u) cos (u) = cos^{2} (u)

cos (u) cos (u)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= cos^{2} (u)

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

\frac{sin ( u )}{1 + \frac{c o s ( u )}{s i n ( u )}} = \frac{sin ^{2} ( u )}{sin ( u ) + cos ( u )}

\frac{sin ( u )}{1 + \frac{c o s ( u )}{s i n ( u )}}

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin (u) = sin (u)

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

= \frac{sin ( u )}{\frac{s i n ( u ) + c o s ( u )}{s i n ( u )}}

Apply the fraction rule:

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

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

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

sin (u) sin (u)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= sin^{2} (u)

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

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

Apply rule

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Cancel the common factor:

cos (u) + sin (u)

= cos (u) - sin (u)

= cos (u) - sin (u)

= cos (u) - sin (u)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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