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

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

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

prove

tan (x) \cdot csc (x) = sec (x)

True

Solution steps

tan (x) csc (x) = sec (x)

Manipulating left side

tan (x) csc (x)

Express with sin, cos

csc (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (θ) sin (θ) = 2 sin^{2} (θ)

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

3 csc (x) cot (x) + 3 csc (x) = 0

sin^{2} (\frac{π}{2}) = 1 - cos^{2} (\frac{π}{2})

cot (θ) - tan (θ) = \frac{2 cos ( 2 θ )}{sin ( 2 θ )}