Is sin(8a)=4sin(2a)cos(2a)cos(4a) ?

The answer to whether sin(8a)=4sin(2a)cos(2a)cos(4a) is True

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prove sin(8a)=4sin(2a)cos(2a)cos(4a)

prove

sin (8 a) = 4 sin (2 a) cos (2 a) cos (4 a)

True

Solution steps

sin (8 a) = 4 sin (2 a) cos (2 a) cos (4 a)

Manipulating right side

4 sin (2 a) cos (2 a) cos (4 a)

Rewrite using trig identities

4 sin (2 a) cos (2 a) cos (4 a)

= 2 \cdot 2 sin (2 a) cos (2 a) cos (4 a)

Use the Double Angle identity:

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

= 2 sin (2 \cdot 2 a) cos (4 a)

Simplify

= 2 sin (4 a) cos (4 a)

Use the Double Angle identity:

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

= sin (2 \cdot 4 a)

Simplify

= sin (8 a)

= sin (8 a)

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (2 x) = \frac{1}{2 ( 1 - cos ^{4} ( x ) )}

\frac{( 1 - sin ( a ) )}{cos ( a )} = cos (a)

csc^{2} (x) (tan (x)) - (tan (x)) = cot (x)

sin^{4} (t) - cos^{4} (t) = 1 - 2 cos^{2} (x)

csc^{2} (A) = 1 + tan^{2} (A)

Is sin(8a)=4sin(2a)cos(2a)cos(4a) ?
The answer to whether sin(8a)=4sin(2a)cos(2a)cos(4a) is True