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

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

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

prove

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

True

Solution steps

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

Manipulating left 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 )}

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

tan (x) tan (x)

Simplify

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

tan (x) tan (x)

Apply exponent rule:

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

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

= tan^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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