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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating right side

csc (2 x)

Express with sin, cos

csc (2 x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

Multiply

2 \cdot \frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )} : \frac{2}{csc ( x ) sec ( x )}

2 \cdot \frac{1}{csc ( x )} \cdot \frac{1}{sec ( 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 2}{csc ( x ) sec ( x )}

Multiply the numbers:

1 \cdot 1 \cdot 2 = 2

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

= \frac{1}{\frac{2}{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 )}{2}

\frac{csc ( x ) sec ( x )}{2}

\frac{csc ( x ) sec ( x )}{2}

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

We showed that the two sides could take the same form

\Rightarrow True

- sec^{2} (x) = - tan^{2} (x) - 1

tan (4 x) \cdot cos (4 x) = sin (4 x)

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

tan (θ) = csc (θ) sec (θ) - cot (θ)

sec (x) - sec (x) sin (x^{2}) = cos (x^{2})

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