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

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

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prove (cot(x)+1)/(cos(x)+sin(x))=csc(x)

prove

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

True

Solution steps

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

Manipulating left side

\frac{cot ( x ) + 1}{cos ( x ) + sin ( 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^{2} (y) = (sec (y))^{2}

csc (A) = \frac{7}{4}

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

csc (cos (+ sin (x))) = cot (+ 1)

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

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