Is cot^2(θ)=cos^2(θ)+(cot(θ)cos(θ))^2 ?

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

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prove cot^2(θ)=cos^2(θ)+(cot(θ)cos(θ))^2

Solution

prove

cot^{2} (θ) = cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Solution

True

Solution steps

cot^{2} (θ) = cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Manipulating right side

cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Express with sin, cos

(cos (θ) cot (θ))^{2} + cos^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

(cos (θ) \frac{cos ( θ )}{sin ( θ )})^{2} + cos^{2} (θ) : \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

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

(cos (θ) \frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

(cos (θ) \frac{cos ( θ )}{sin ( θ )})^{2}

Multiply

cos (θ) \frac{cos ( θ )}{sin ( θ )} : \frac{cos ^{2} ( θ )}{sin ( θ )}

cos (θ) \frac{cos ( θ )}{sin ( θ )}

Multiply fractions:

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

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{cos ^{2} ( θ )}{sin ( θ )}

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

Apply exponent rule:

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

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

(cos^{2} (θ))^{2} : cos^{4} (θ)

Apply exponent rule:

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

= cos^{2 \cdot 2} (θ)

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (θ)

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )} + cos^{2} (θ)

Convert element to fraction:

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

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )} + \frac{cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Factor

\frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )} : \frac{cos ^{2} ( θ ) ( cos ^{2} ( θ ) + sin ^{2} ( θ ) )}{sin ^{2} ( θ )}

\frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Factor

cos^{4} (θ) + cos^{2} (θ) sin^{2} (θ) : cos^{2} (θ) (cos^{2} (θ) + sin^{2} (θ))

cos^{4} (θ) + cos^{2} (θ) sin^{2} (θ)

Apply exponent rule:

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

cos^{4} (θ) = cos^{2} (θ) cos^{2} (θ)

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

Factor out common term

cos^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

\frac{cos ( θ ) cot ( θ )}{sin ( θ )}

= cot (θ) \frac{cos ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

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

cot (θ) cot (θ)

Simplify

cot (θ) cot (θ) : cot^{2} (θ)

cot (θ) cot (θ)

Apply exponent rule:

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

cot (θ) cot (θ) = cot^{1 + 1} (θ)

= cot^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cot^{2} (θ)

cot^{2} (θ)

cot^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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