Is tan(x)cos(x)-1/(csc(x))=0 ?

The answer to whether tan(x)cos(x)-1/(csc(x))=0 is True

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prove tan(x)cos(x)-1/(csc(x))=0

prove

tan (x) cos (x) - \frac{1}{csc ( x )} = 0

True

Solution steps

tan (x) cos (x) - \frac{1}{csc ( x )} = 0

Manipulating left side

tan (x) cos (x) - \frac{1}{csc ( x )}

Express with sin, cos

- \frac{1}{csc ( x )} + cos (x) tan (x)

Use the basic trigonometric identity:

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

= - \frac{1}{\frac{1}{s i n ( x )}} + cos (x) tan (x)

Use the basic trigonometric identity:

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

= - \frac{1}{\frac{1}{s i n ( x )}} + cos (x) \frac{sin ( x )}{cos ( x )}

- \frac{1}{\frac{1}{s i n ( x )}} + cos (x) \frac{sin ( x )}{cos ( x )} = 0

- \frac{1}{\frac{1}{s i n ( x )}} + cos (x) \frac{sin ( x )}{cos ( x )}

\frac{1}{\frac{1}{s i n ( x )}} = sin (x)

\frac{1}{\frac{1}{s i n ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sin (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= - sin (x) + sin (x)

Add similar elements:

- sin (x) + sin (x) = 0

= 0

= 0

= 0

We showed that the two sides could take the same form

\Rightarrow True

sec (a) - cos (a) = sin (a) tan (a)

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

sin^{7} (x) = (1 - cos^{2} (x))^{3} sin (x)

tan (A) + cot (A) = sec (A) csc (A)

\frac{sec ( θ )}{csc ( θ )} = tan (θ)

Is tan(x)cos(x)-1/(csc(x))=0 ?
The answer to whether tan(x)cos(x)-1/(csc(x))=0 is True