Is 2sin^3(x)cos(x)+2sin(x)cos^3(x)=sin(2x) ?

The answer to whether 2sin^3(x)cos(x)+2sin(x)cos^3(x)=sin(2x) is True

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prove 2sin^3(x)cos(x)+2sin(x)cos^3(x)=sin(2x)

prove

2 sin^{3} (x) cos (x) + 2 sin (x) cos^{3} (x) = sin (2 x)

True

Solution steps

2 sin^{3} (x) cos (x) + 2 sin (x) cos^{3} (x) = sin (2 x)

Manipulating left side

2 sin^{3} (x) cos (x) + 2 sin (x) cos^{3} (x)

Factor

2 cos (x) sin^{3} (x) + 2 cos^{3} (x) sin (x) : 2 sin (x) cos (x) (sin^{2} (x) + cos^{2} (x))

2 cos (x) sin^{3} (x) + 2 cos^{3} (x) sin (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{3} (x) = sin (x) sin^{2} (x), cos^{3} (x) = cos (x) cos^{2} (x)

= 2 cos (x) sin (x) sin^{2} (x) + 2 cos (x) cos^{2} (x) sin (x)

Rewrite as

= 2 sin (x) cos (x) sin^{2} (x) + 2 sin (x) cos (x) cos^{2} (x)

Factor out common term

2 sin (x) cos (x)

= 2 sin (x) cos (x) (sin^{2} (x) + cos^{2} (x))

= (cos^{2} (x) + sin^{2} (x)) \cdot 2 cos (x) sin (x)

Rewrite using trig identities

(cos^{2} (x) + sin^{2} (x)) \cdot 2 cos (x) sin (x)

Use the Double Angle identity:

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

= sin (2 x) (cos^{2} (x) + sin^{2} (x))

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= sin (2 x) \cdot 1

Simplify

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

Is 2sin^3(x)cos(x)+2sin(x)cos^3(x)=sin(2x) ?
The answer to whether 2sin^3(x)cos(x)+2sin(x)cos^3(x)=sin(2x) is True