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

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

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prove csc(a)sec(a)=2csc(2a)

prove

csc (a) sec (a) = 2 csc (2 a)

True

Solution steps

csc (a) sec (a) = 2 csc (2 a)

Manipulating left side

csc (a) sec (a)

Express with sin, cos

csc (a) sec (a)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( a )} sec (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

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

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

Rewrite using trig identities

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

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 a )}{2}}

Apply the fraction rule:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 2 csc (2 a)

2 csc (2 a)

2 csc (2 a)

We showed that the two sides could take the same form

\Rightarrow True