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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

cos^{2} (α) + (cot (α) cos (α))^{2}

Express with sin, cos

(cos (α) cot (α))^{2} + cos^{2} (α)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply

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

cos (α) \frac{cos ( α )}{sin ( α )}

Multiply fractions:

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

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

cos (α) cos (α) = cos^{2} (α)

cos (α) cos (α)

Apply exponent rule:

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

cos (α) cos (α) = cos^{1 + 1} (α)

= cos^{1 + 1} (α)

Add the numbers:

1 + 1 = 2

= cos^{2} (α)

= \frac{cos ^{2} ( α )}{sin ( α )}

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

Apply exponent rule:

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

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

(cos^{2} (α))^{2} : cos^{4} (α)

Apply exponent rule:

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

= cos^{2 \cdot 2} (α)

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (α)

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

Factor

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

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

Factor

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

cos^{4} (α) + cos^{2} (α) sin^{2} (α)

Apply exponent rule:

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

cos^{4} (α) = cos^{2} (α) cos^{2} (α)

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

Factor out common term

cos^{2} (α)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

\frac{cos ( α ) cot ( α )}{sin ( α )}

= cot (α) \frac{cos ( α )}{sin ( α )}

Use the basic trigonometric identity:

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

cot (α) cot (α)

Simplify

cot (α) cot (α) : cot^{2} (α)

cot (α) cot (α)

Apply exponent rule:

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

cot (α) cot (α) = cot^{1 + 1} (α)

= cot^{1 + 1} (α)

Add the numbers:

1 + 1 = 2

= cot^{2} (α)

cot^{2} (α)

cot^{2} (α)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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