Is (tan(x)csc(x))/(sec(x))=1 ?

The answer to whether (tan(x)csc(x))/(sec(x))=1 is True

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prove (tan(x)csc(x))/(sec(x))=1

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( x )} tan ( x )}{sec ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{sec ( x )}

Use the basic trigonometric identity:

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

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

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

Cancel the common factor:

cos (x)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( 3 x ) - sin ( 5 x )}{2 cos ( 4 x )} = - sin (x)

2 tan (θ) (1 + cos (2 θ)) = 2 sin (2 θ)

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

tan (a) + cot (a) = 2 csc (2 a)

\frac{sec ^{2} ( t )}{csc ( t )} = sec (t) tan (t)

Is (tan(x)csc(x))/(sec(x))=1 ?
The answer to whether (tan(x)csc(x))/(sec(x))=1 is True