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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (x) + sin (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

cos (x) sec (x) = 1

\frac{sec ( t ) - cos ( t )}{sec ( t )} = sin^{2} (t)

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

sin (\frac{π}{2} - x) = sin (\frac{π}{2} + x)

\frac{sin ^{2} ( t )}{cos ( t )} = sec (t) - cos (t)

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