Is cot(x)-csc^2(x)cot(x)=-cot^3(x) ?

The answer to whether cot(x)-csc^2(x)cot(x)=-cot^3(x) is True

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prove cot(x)-csc^2(x)cot(x)=-cot^3(x)

prove

cot (x) - csc^{2} (x) cot (x) = - cot^{3} (x)

True

Solution steps

cot (x) - csc^{2} (x) cot (x) = - cot^{3} (x)

Manipulating left side

cot (x) - csc^{2} (x) cot (x)

Rewrite using trig identities

cot (x) - csc^{2} (x) cot (x)

Use the Pythagorean identity:

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

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

Simplify

cot (x) - (1 + cot^{2} (x)) cot (x) : - cot^{3} (x)

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

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

Expand

- cot (x) (1 + cot^{2} (x)) : - cot (x) - cot^{3} (x)

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

Apply the distributive law:

a (b + c) = ab + a c

a = - cot (x), b = 1, c = cot^{2} (x)

= - cot (x) \cdot 1 + (- cot (x)) cot^{2} (x)

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot cot (x) - cot^{2} (x) cot (x)

Simplify

- 1 \cdot cot (x) - cot^{2} (x) cot (x) : - cot (x) - cot^{3} (x)

- 1 \cdot cot (x) - cot^{2} (x) cot (x)

1 \cdot cot (x) = cot (x)

1 \cdot cot (x)

Multiply:

1 \cdot cot (x) = cot (x)

= cot (x)

cot^{2} (x) cot (x) = cot^{3} (x)

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

Apply exponent rule:

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

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

= cot^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cot^{3} (x)

= - cot (x) - cot^{3} (x)

= - cot (x) - cot^{3} (x)

= cot (x) - cot (x) - cot^{3} (x)

Add similar elements:

cot (x) - cot (x) = 0

= - cot^{3} (x)

= - cot^{3} (x)

= - cot^{3} (x)

We showed that the two sides could take the same form

\Rightarrow True

cot (x - \frac{π}{2}) = - tan (x)

\frac{sin ( 5 x ) + sin ( 11 x )}{cos ( 5 x ) + cos ( 11 x )} = tan (8 x)

\frac{sec ( x ) - 1}{sin ( x ) sec ( x )} = tan (\frac{x}{2})

cos (4 x) - sin (4 x) = cos (2 x)

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

Is cot(x)-csc^2(x)cot(x)=-cot^3(x) ?
The answer to whether cot(x)-csc^2(x)cot(x)=-cot^3(x) is True