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

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

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

prove

sec (x) sin (2 x) = 2 sin (x)

True

Solution steps

sec (x) sin (2 x) = 2 sin (x)

Manipulating left side

sec (x) sin (2 x)

Express with sin, cos

sec (x) sin (2 x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Cancel the common factor:

cos (x)

= 2 sin (x)

= 2 sin (x)

We showed that the two sides could take the same form

\Rightarrow True

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cos (x) \cdot cot (x) = csc (x) - sin (x)

cos (x + 27 0^{\circ}) = sin (x)

cos (u) sec (u) - sin^{2} (u) = cos^{2} (u)

cos^{4} (x) = cos^{2} (x) cos^{2} (x)