Is (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x) ?

The answer to whether (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x) is True

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prove (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x)

prove

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )} = 2 tan (x)

True

Solution steps

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )} = 2 tan (x)

Manipulating left side

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )}

Express with sin, cos

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

\frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}} + \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

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

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

Divide fractions:

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

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

Refine

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

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

Apply rule

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

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

Add similar elements:

sin (x) + sin (x) = 2 sin (x)

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

Rewrite using trig identities

= 2 \cdot \frac{sin ( x )}{cos ( x )}

Use the basic trigonometric identity:

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

2 tan (x)

2 tan (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sec (2 θ) = \frac{sec ^{2} ( θ )}{1 - tan ^{2} ( θ )}

cos (2 a) = cos^{2} (a) - sin^{2} (a)

tan (θ) + cot (θ) = sec (θ) \cdot csc (θ)

\frac{cos ( x )}{csc ( x ) + 1} + \frac{cos ( x )}{csc ( x ) - 1} = 2 tan (x)

Is (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x) ?
The answer to whether (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x) is True