Is (cos(u)sec(u))/(tan(u))=cot(u) ?

The answer to whether (cos(u)sec(u))/(tan(u))=cot(u) is True

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prove (cos(u)sec(u))/(tan(u))=cot(u)

prove

\frac{cos ( u ) sec ( u )}{tan ( u )} = cot (u)

True

Solution steps

\frac{cos ( u ) sec ( u )}{tan ( u )} = cot (u)

Manipulating left side

\frac{cos ( u ) sec ( u )}{tan ( u )}

Express with sin, cos

\frac{cos ( u ) sec ( u )}{tan ( u )}

Use the basic trigonometric identity:

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

= \frac{cos ( u ) \frac{1}{c o s ( u )}}{tan ( u )}

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{cos ( u ) \frac{1}{c o s ( u )}}{\frac{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 ) \frac{1}{c o s ( u )} cos ( u )}{sin ( u )}

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

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

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

Multiply

\frac{1}{cos ( u )} cos^{2} (u) : cos (u)

\frac{1}{cos ( u )} cos^{2} (u)

Multiply fractions:

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

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

Multiply:

1 \cdot cos^{2} (u) = cos^{2} (u)

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

Cancel the common factor:

cos (u)

= cos (u)

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

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

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

Use the basic trigonometric identity:

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

= cot (u)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e sin (x) (1 + cot^{2} (x)) = csc (x)

p ro v e sin^{4} (x) - cos^{4} (x) = 1 - 2 cos^{2} (x)

p ro v e sec^{4} (x) - tan^{4} (x) = 1 + 2 tan^{2} (x)

p ro v e sin (- x) = - sin (x)

p ro v e sin (x + π) = - sin (x)

Is (cos(u)sec(u))/(tan(u))=cot(u) ?
The answer to whether (cos(u)sec(u))/(tan(u))=cot(u) is True