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

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

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

prove

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

True

Solution steps

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

Manipulating left side

tan (A)

Express with sin, cos

tan (A)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

Divide fractions:

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

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

Refine

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

csc (- x) = - csc (x)

sin (β) + cos (β) cot (β) = csc (β)

cos (\frac{3 π}{2} + x) = - cos (x)

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

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

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