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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{1 + \frac{1}{c o s ( x )}}{sin ( x ) + tan ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

Divide fractions:

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

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

Cancel the common factor:

cos (x)

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

Factor out common term

sin (x)

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

Cancel the common factor:

cos (x) + 1

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

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

= \frac{1}{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 )}}

Simplify

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Apply rule

\frac{a}{1} = a

= csc (x)

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