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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

1 + \frac{sin ( θ )}{cos ( θ )}

Convert element to fraction:

1 = \frac{1 cos ( θ )}{cos ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (θ) = cos (θ)

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (θ) + sin (θ)

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

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

tan (x) \cdot sin (x) + cos (x) = sec (x)

cos (θ) + cos (θ) tan^{2} (θ) = sec (θ)

\frac{cos ( θ ) + sin ( θ )}{sin ( θ )} = 1 + cot (θ)

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