Is (sin(x)csc(x))/(cos(x))=sec(x) ?

The answer to whether (sin(x)csc(x))/(cos(x))=sec(x) is True

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prove (sin(x)csc(x))/(cos(x))=sec(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (x) + sec (x) = 1

- sinh^{2} (x) + cosh^{2} (x) = 1

sin^{2} (x) + cos^{2} (x) - 1 = 0

sec (x) sin (\frac{π}{2} - x) = 1

csc (x) tan^{2} (x) = \frac{sin ( x )}{cos ^{2} ( x )}

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