Is sin^2(u)(cot^2(u)+1)=1 ?

The answer to whether sin^2(u)(cot^2(u)+1)=1 is True

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prove sin^2(u)(cot^2(u)+1)=1

prove

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

True

Solution steps

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

Manipulating left side

sin^{2} (u) (cot^{2} (u) + 1)

Express with sin, cos

(1 + cot^{2} (u)) sin^{2} (u)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (u)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

Is sin^2(u)(cot^2(u)+1)=1 ?
The answer to whether sin^2(u)(cot^2(u)+1)=1 is True