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

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

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prove csc(s)=cot(s)sec(s)

prove

csc (s) = cot (s) sec (s)

True

Solution steps

csc (s) = cot (s) sec (s)

Manipulating right side

cot (s) sec (s)

Express with sin, cos

cot (s) sec (s)

Use the basic trigonometric identity:

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

= \frac{cos ( s )}{sin ( s )} sec (s)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (s)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (s)

csc (s)

csc (s)

We showed that the two sides could take the same form

\Rightarrow True

csc^{4} (x) - cot^{4} (x) = 2 sec^{2} (x) - 1

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

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

tan^{2} (x) = (sec (x) - 1) (sec (x) + 1)

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