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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

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 )}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= cos (x) + sin (x)

= cos (x) + sin (x)

= cos (x) + sin (x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{sin ( 4 x )}{4} = \frac{sin ( x ) cos ( x )}{2}

p ro v e \frac{sin ( x ) tan ( x )}{cos ( x ) + 1} = sec (x) - 1

p ro v e sin (a + b) - sin (a - b) = 2 sin (a) sin (b)

p ro v e sec (x) + 1 = \frac{tan ^{2} ( x )}{sec ( x ) - 1}

p ro v e \frac{sin ( x )}{1 - cos ^{2} ( x )} = cos (x)

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