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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (x)

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

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

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

Rewrite using trig identities

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

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

cot (\frac{x}{2}) = \frac{( 1 + cos ( x ) )}{sin ( x )}

\frac{1}{tan ( b )} + tan (b) = sec (b) csc (b)

csc^{2} (x) cot^{2} (x) + csc^{2} (x) = csc^{4} (x)

sin^{5} (x) = (sin (x))^{5}

2 tan (x) sec (x) csc (x) = 2 + 2 tan^{2} (x)

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