Is csc(x)tan(x)cos(x)=1 ?

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

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prove csc(x)tan(x)cos(x)=1

prove

csc (x) tan (x) cos (x) = 1

True

Solution steps

csc (x) tan (x) cos (x) = 1

Manipulating left side

csc (x) tan (x) cos (x)

Express with sin, cos

cos (x) csc (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

Cancel the common factor:

cos (x)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 5 = 3 tan (\frac{x}{2})

\frac{sin ( x )}{cos ^{2} ( x )} = tan (x) sec (x)

\frac{cot ( x ) + 2}{csc ( x )} = 2 sin (x) + cos (x)

(sec (x)) (sin (x)) = tan (x)

tan (2 x) = 2 sin (x) cos (x) sec (2 x)