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

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

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Popular Trigonometry >

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

\frac{sec ^{2} ( x ) cot ( x )}{csc ^{2} ( x )}

Express with sin, cos

\frac{cot ( x ) sec ^{2} ( x )}{csc ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply

\frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2} : \frac{1}{cos ( x ) sin ( x )}

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

Multiply fractions:

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

= \frac{cos ( x ) ( \frac{1}{c o s ( x )} ) ^{2}}{sin ( x )}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{c o s ^{2} ( x )} cos ( x )}{sin ( x )}

Multiply

cos (x) \frac{1}{cos ^{2} ( x )} : \frac{1}{cos ( x )}

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

Cancel the common factor:

cos (x)

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

= \frac{\frac{1}{c o s ( x )}}{sin ( x )}

Apply the fraction rule:

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

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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