Is (csc(z))/(sec(z))=cot(z) ?

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

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prove (csc(z))/(sec(z))=cot(z)

prove

\frac{csc ( z )}{sec ( z )} = cot (z)

True

Solution steps

\frac{csc ( z )}{sec ( z )} = cot (z)

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( z )}}{sec ( z )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( z )}}{\frac{1}{c o s ( z )}}

Simplify

\frac{\frac{1}{s i n ( z )}}{\frac{1}{c o s ( z )}} : \frac{cos ( z )}{sin ( z )}

\frac{\frac{1}{s i n ( z )}}{\frac{1}{c o s ( z )}}

Divide fractions:

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

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

Refine

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

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

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

Use the basic trigonometric identity:

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

= cot (z)

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( θ ) cot ( θ )}{sec ( θ )} = cos (θ)

tan^{2} (b) - tan^{2} (b) sin^{2} (b) = sin^{2} (b)

2 cos (x) sin (x) = sin (2 x)

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

sin (x) + tan (x) = tan (x) (1 + cos (x))

Is (csc(z))/(sec(z))=cot(z) ?
The answer to whether (csc(z))/(sec(z))=cot(z) is True