Is (sin(x)csc(x))/(cot(x))=tan(x) ?

The answer to whether (sin(x)csc(x))/(cot(x))=tan(x) is True

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prove (sin(x)csc(x))/(cot(x))=tan(x)

prove

\frac{sin ( x ) csc ( x )}{cot ( x )} = tan (x)

True

Solution steps

\frac{sin ( x ) csc ( x )}{cot ( x )} = tan (x)

Manipulating left side

\frac{sin ( x ) csc ( x )}{cot ( x )}

Express with sin, cos

\frac{csc ( x ) sin ( x )}{cot ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

\frac{1}{sin ( x )} sin (x) sin (x) = \frac{1}{sin ( x )} sin^{2} (x)

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

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

Add the numbers:

1 + 1 = 2

= \frac{1}{sin ( x )} sin^{2} (x)

= \frac{\frac{1}{s i n ( x )} sin ^{2} ( x )}{cos ( x )}

Multiply

\frac{1}{sin ( x )} sin^{2} (x) : sin (x)

\frac{1}{sin ( x )} sin^{2} (x)

Multiply fractions:

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

= \frac{1 \cdot sin ^{2} ( x )}{sin ( x )}

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (θ) (tan^{2} (θ) + 1) = 1

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

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

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

sin (t) csc (t) = 1

Is (sin(x)csc(x))/(cot(x))=tan(x) ?
The answer to whether (sin(x)csc(x))/(cot(x))=tan(x) is True