Is cot(x)(sin(x)+tan(x))=1+cos(x) ?

The answer to whether cot(x)(sin(x)+tan(x))=1+cos(x) is True

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

prove

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

True

Solution steps

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

Manipulating left side

cot (x) (sin (x) + tan (x))

Express with sin, cos

(sin (x) + tan (x)) cot (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x) cos (x) + sin (x)

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

Factor out common term

sin (x)

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

Cancel the common factor:

sin (x)

= cos (x) + 1

= cos (x) + 1

= 1 + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

csc^{2} (x) (1 - cos^{2} (x)) = tan (42 0^{\circ})

1 - cot (x) = 1

cot (x) - csc (x) = - \frac{1}{cot ( x ) + csc ( x )}

cos (θ + 3 0^{\circ}) - sin (θ + 6 0^{\circ}) = - sin (θ)

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

Is cot(x)(sin(x)+tan(x))=1+cos(x) ?
The answer to whether cot(x)(sin(x)+tan(x))=1+cos(x) is True