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

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

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

prove

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

True

Solution steps

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

Manipulating right side

1 + tan (θ)

Express with sin, cos

1 + tan (θ)

Use the basic trigonometric identity:

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

= 1 + \frac{sin ( θ )}{cos ( θ )}

Simplify

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{cos ( θ ) + sin ( θ )}{cos ( θ )}

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

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

We showed that the two sides could take the same form

\Rightarrow True

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