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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Multiply fractions:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

cot (x) cot (x)

Simplify

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

Apply exponent rule:

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

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

= cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (\frac{π}{6} + x) = cos (\frac{π}{3} - x)

2 cos^{2} (x) - 1 = 0

(1 + cos (2 θ)) (1 - cos (2 θ)) = sin^{2} (2 θ)

sin (x) - cos (x) = 0

sin (2 θ) + cos (2 θ) = 2 sin (θ) cos (θ) + 2 cos^{2} (θ) - 1

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