Is tan(s)csc(s)=sec(s) ?

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

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prove tan(s)csc(s)=sec(s)

prove

tan (s) csc (s) = sec (s)

True

Solution steps

tan (s) csc (s) = sec (s)

Manipulating left side

tan (s) csc (s)

Express with sin, cos

csc (s) tan (s)

Use the basic trigonometric identity:

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

= \frac{1}{sin ( s )} tan (s)

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{1}{sin ( s )} \cdot \frac{sin ( s )}{cos ( s )}

Multiply fractions:

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

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

Cancel the common factor:

sin (s)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (s)

sec (s)

sec (s)

We showed that the two sides could take the same form

\Rightarrow True

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tan (\frac{3 π}{2} - x) = - tan (x)

cos (2 α) = cos (α + α)

cos^{4} (x) - sin^{4} (x) = 2 tan^{2} (x) - 1

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