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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

csc (θ) + sec (θ) = tan (θ) + 1

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

\frac{cot ( θ ) - tan ( θ )}{cot ( θ ) + tan ( θ )} = cos (2 θ)

cos^{2} (2 x) + 4 sin^{2} (x) cos^{2} (x) = 1

cos (x) (sec (x) - 1) = 1 - cos (x)

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