Is (1+csc(x))/(sec(x))=cos(x)+cot(x) ?

The answer to whether (1+csc(x))/(sec(x))=cos(x)+cot(x) is True

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prove (1+csc(x))/(sec(x))=cos(x)+cot(x)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

cos (x) + cot (x)

Express with sin, cos

cos (x) + cot (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

Factor

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

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

Factor

cos (x) + cos (x) sin (x) : cos (x) (1 + sin (x))

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

Factor out common term

cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{( 1 + \frac{1}{c s c ( x )} ) cos ( x )}{\frac{1}{c s c ( x )}}

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

= \frac{( 1 + \frac{1}{c s c ( x )} ) \frac{1}{s e c ( x )} csc ( x )}{1}

Join

1 + \frac{1}{csc ( x )} : \frac{csc ( x ) + 1}{csc ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot csc (x) = csc (x)

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

= \frac{\frac{c s c ( x ) + 1}{c s c ( x )} \cdot \frac{1}{s e c ( x )} csc ( x )}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{csc ( x ) + 1}{csc ( x )} \cdot \frac{1}{sec ( x )} csc (x)

Multiply fractions:

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

= \frac{( csc ( x ) + 1 ) \cdot 1 \cdot csc ( x )}{csc ( x ) sec ( x )}

Cancel the common factor:

csc (x)

= \frac{( csc ( x ) + 1 ) \cdot 1}{sec ( x )}

(csc (x) + 1) \cdot 1 = csc (x) + 1

(csc (x) + 1) \cdot 1

Multiply:

(csc (x) + 1) \cdot 1 = (csc (x) + 1)

= (csc (x) + 1)

Remove parentheses:

(a) = a

= csc (x) + 1

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

\frac{csc ( x ) + 1}{sec ( x )}

\frac{csc ( x ) + 1}{sec ( x )}

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (1+csc(x))/(sec(x))=cos(x)+cot(x) ?
The answer to whether (1+csc(x))/(sec(x))=cos(x)+cot(x) is True