Is tan(2x)=2sin(x)cos(x)sec(2x) ?

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

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prove tan(2x)=2sin(x)cos(x)sec(2x)

prove

tan (2 x) = 2 sin (x) cos (x) sec (2 x)

True

Solution steps

tan (2 x) = 2 sin (x) cos (x) sec (2 x)

Manipulating right side

2 sin (x) cos (x) sec (2 x)

Rewrite using trig identities

2 sin (x) cos (x) sec (2 x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= sin (2 x) sec (2 x)

Express with sin, cos

sec (2 x) sin (2 x)

Use the basic trigonometric identity:

sec (2 x) = \frac{1}{cos ( 2 x )}

= \frac{1}{cos ( 2 x )} sin (2 x)

Simplify

\frac{1}{cos ( 2 x )} sin (2 x) : \frac{sin ( 2 x )}{cos ( 2 x )}

\frac{1}{cos ( 2 x )} sin (2 x)

Multiply fractions:

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

= \frac{1 sin ( 2 x )}{cos ( 2 x )}

Multiply:

1 \cdot sin (2 x) = sin (2 x)

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

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

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

Use the basic trigonometric identity:

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

= tan (2 x)

= tan (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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