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

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

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

prove

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

True

Solution steps

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

Manipulating left side

2 sec (x) cot (x)

Express with sin, cos

2 cot (x) sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 2 csc (x)

2 csc (x)

2 csc (x)

We showed that the two sides could take the same form

\Rightarrow True

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