Is (cot(x)+1)/(sin(x)+cos(x))=csc(x) ?

The answer to whether (cot(x)+1)/(sin(x)+cos(x))=csc(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)+1)/(sin(x)+cos(x))=csc(x)

prove

\frac{cot ( x ) + 1}{sin ( x ) + cos ( x )} = csc (x)

True

Solution steps

\frac{cot ( x ) + 1}{sin ( x ) + cos ( x )} = csc (x)

Manipulating left side

\frac{cot ( x ) + 1}{sin ( x ) + cos ( x )}

Express with sin, cos

\frac{1 + cot ( x )}{cos ( x ) + sin ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

sin (x) + cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

We showed that the two sides could take the same form

\Rightarrow True

sec (A) csc (A) = tan (A) cot (A)

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

tan^{4} (θ) - sec^{4} (θ) = 1 - 2 sec^{2} (θ)

\frac{( - sin ^{2} ( a ) + 1 )}{cos ( a )} = cos (a)

2 csc (2 x) tan (x) = sec^{2} (x)

Is (cot(x)+1)/(sin(x)+cos(x))=csc(x) ?
The answer to whether (cot(x)+1)/(sin(x)+cos(x))=csc(x) is True