Is 2csc(2x)=csc(x)sec(x) ?

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

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prove 2csc(2x)=csc(x)sec(x)

prove

2 csc (2 x) = csc (x) sec (x)

True

Solution steps

2 csc (2 x) = csc (x) sec (x)

Manipulating left side

2 csc (2 x)

Express with sin, cos

2 csc (2 x)

Use the basic trigonometric identity:

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

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

Simplify

2 \cdot \frac{1}{sin ( 2 x )} : \frac{2}{sin ( 2 x )}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Divide the numbers:

\frac{2}{2} = 1

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Simplify

\frac{1}{\frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Multiply

\frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )} : \frac{1}{csc ( x ) sec ( x )}

\frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )}

Multiply fractions:

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

= \frac{1 \cdot 1}{csc ( x ) sec ( x )}

Multiply the numbers:

1 \cdot 1 = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (x) sec (x)

csc (x) sec (x)

csc (x) sec (x)

We showed that the two sides could take the same form

\Rightarrow True

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