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

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

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

prove

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

True

Solution steps

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

Manipulating right side

2 sin (t) cos (t) sec (2 t)

Rewrite using trig identities

2 sin (t) cos (t) sec (2 t)

Use the Double Angle identity:

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

= sin (2 t) sec (2 t)

Express with sin, cos

sec (2 t) sin (2 t)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

Use the basic trigonometric identity:

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

= tan (2 t)

= tan (2 t)

We showed that the two sides could take the same form

\Rightarrow True

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