Is sin(3x)=2*sin((3x)/2)*cos((3x)/2) ?

The answer to whether sin(3x)=2*sin((3x)/2)*cos((3x)/2) is True

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prove sin(3x)=2sin((3x)/2)cos((3x)/2)

prove

sin (3 x) = 2 \cdot sin (\frac{3 x}{2}) \cdot cos (\frac{3 x}{2})

True

Solution steps

sin (3 x) = 2 sin (\frac{3 x}{2}) cos (\frac{3 x}{2})

Let:

u = \frac{3 x}{2}

sin (2 u) = 2 sin (u) cos (u)

Prove

sin (2 u) = 2 sin (u) cos (u) :

True

sin (2 u) = 2 sin (u) cos (u)

Manipulating left side

sin (2 u)

Rewrite using trig identities

sin (2 u)

Rewrite as

= sin (u + u)

Use the Angle Sum identity:

sin (s + s) = sin (s) cos (s) + cos (s) sin (s)

= sin (u) cos (u) + cos (u) sin (u)

= sin (u) cos (u) + cos (u) sin (u)

Simplify

sin (u) cos (u) + cos (u) sin (u) : 2 sin (u) cos (u)

sin (u) cos (u) + cos (u) sin (u)

Add similar elements:

sin (u) cos (u) + cos (u) sin (u) = 2 sin (u) cos (u)

= 2 sin (u) cos (u)

= 2 sin (u) cos (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

sin (3 x) = 2 sin (\frac{3 x}{2}) cos (\frac{3 x}{2})

We showed that the two sides could take the same form

\Rightarrow True

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(sec (a) - tan (a)) (csc (a) + 1) = cot (a)

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

(cot (x) + tan (x))^{2} = sec^{2} (x) csc^{2} (x)

6 sin (\frac{5 π}{6}) = 2 + 2 sin (\frac{5 π}{6})

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