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

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

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

prove

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

True

Solution steps

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

Manipulating left side

tan (x)

Express with sin, cos

tan (x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

Divide fractions:

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

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

Refine

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( θ )}{cot ( θ )} = sec (θ)

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

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

coth^{2} (x) - csch^{2} (x) = 1

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

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