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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( x )} sec ( x )}{cot ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )}}{cot ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )} sin ( x )}{cos ( x )}

Multiply

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} sin (x) : \frac{1}{cos ( x )}

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

Multiply the numbers:

1 \cdot 1 = 1

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{1}{\frac{1}{s e c ^{2} ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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