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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

Apply the fraction rule:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

= \frac{2 csc ( 2 x )}{1}

Apply rule

\frac{a}{1} = a

= 2 csc (2 x)

2 csc (2 x)

2 csc (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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