Is (cot(x)+2)/(csc(x))=2sin(x)+cos(x) ?

The answer to whether (cot(x)+2)/(csc(x))=2sin(x)+cos(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (cot(x)+2)/(csc(x))=2sin(x)+cos(x)

prove

\frac{cot ( x ) + 2}{csc ( x )} = 2 sin (x) + cos (x)

True

Solution steps

\frac{cot ( x ) + 2}{csc ( x )} = 2 sin (x) + cos (x)

Manipulating left side

\frac{cot ( x ) + 2}{csc ( x )}

Express with sin, cos

\frac{2 + cot ( x )}{csc ( x )}

Use the basic trigonometric identity:

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

= \frac{2 + \frac{c o s ( x )}{s i n ( x )}}{csc ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Join

2 + \frac{cos ( x )}{sin ( x )} : \frac{2 sin ( x ) + cos ( x )}{sin ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 2 sin (x) + cos (x)

= 2 sin (x) + cos (x)

= cos (x) + 2 sin (x)

= 2 sin (x) + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

(sec (x)) (sin (x)) = tan (x)

tan (2 x) = 2 sin (x) cos (x) sec (2 x)

sin^{3} (z) cos^{2} (z) = sin^{3} (z) - sin^{5} (z)

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

sec (x) = sin (x) \cdot (tan (x) + cot (x))

Is (cot(x)+2)/(csc(x))=2sin(x)+cos(x) ?
The answer to whether (cot(x)+2)/(csc(x))=2sin(x)+cos(x) is True