Is (1+sec(x))/(csc(x))=sin(x)+tan(x) ?

The answer to whether (1+sec(x))/(csc(x))=sin(x)+tan(x) is True

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prove (1+sec(x))/(csc(x))=sin(x)+tan(x)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

sin (x) + tan (x)

Express with sin, cos

sin (x) + tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Factor

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

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

Factor

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

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

Factor out common term

sin (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sec (x) = sec (x)

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

sec (x)

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

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

(sec (x) + 1) \cdot 1

Multiply:

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

= (sec (x) + 1)

Remove parentheses:

(a) = a

= sec (x) + 1

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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