Is (tan(x)cot(x))/(sin(x))=csc(x) ?

The answer to whether (tan(x)cot(x))/(sin(x))=csc(x) is True

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prove (tan(x)cot(x))/(sin(x))=csc(x)

prove

\frac{tan ( x ) cot ( x )}{sin ( x )} = csc (x)

True

Solution steps

\frac{tan ( x ) cot ( x )}{sin ( x )} = csc (x)

Manipulating left side

\frac{tan ( x ) cot ( x )}{sin ( x )}

Express with sin, cos

\frac{cot ( x ) tan ( x )}{sin ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{sin ( x )}

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= 1

= \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

csc^{2} (\frac{θ}{2}) = \frac{2 sec ( θ )}{sec ( θ ) - 1}

csc^{2} (x) \cdot tan^{2} (x) - 1 = tan^{2} (x)

\frac{1 + csc ( θ )}{sec ( θ )} = cos (θ) + cot (θ)

\frac{cos ( x )}{sec ( 2 x )} - \frac{sin ( x )}{csc ( 2 x )} = cos (3 x)

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

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