Is 1/(cot^2(x))=(tan(x))/(cot(x)) ?

The answer to whether 1/(cot^2(x))=(tan(x))/(cot(x)) is True

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prove 1/(cot^2(x))=(tan(x))/(cot(x))

prove

\frac{1}{cot ^{2} ( x )} = \frac{tan ( x )}{cot ( x )}

True

Solution steps

\frac{1}{cot ^{2} ( x )} = \frac{tan ( x )}{cot ( x )}

Manipulating left side

\frac{1}{cot ^{2} ( x )}

Express with sin, cos

\frac{1}{cot ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{1}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

Simplify

\frac{1}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{1}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

= \frac{1}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

Manipulating right side

\frac{tan ( x )}{cot ( x )}

Express with sin, cos

\frac{tan ( x )}{cot ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{s i n ( x )}{c o s ( x )}}{cot ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )}}

Simplify

\frac{\frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )}} : \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{\frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ( x )}{s i n ( x )}}

Divide fractions:

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

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

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

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (x) + sin (x) csc (x) = sec^{2} (x)

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

\frac{sin ( x )}{cos ( x )} = \frac{1}{cot ( x )}

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

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

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