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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

(5 sin (θ) - 5 sin (θ))^{2} = 25 - 25 sin (2 θ)

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

\frac{1}{( sec ( y ) - tan ( y ) )} = sec (y) + tan (y)

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