Is 2sin^2(x/2)+4cos^2(x/2)-3=cos(x) ?

The answer to whether 2sin^2(x/2)+4cos^2(x/2)-3=cos(x) is True

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prove 2sin^2(x/2)+4cos^2(x/2)-3=cos(x)

prove

2 sin^{2} (\frac{x}{2}) + 4 cos^{2} (\frac{x}{2}) - 3 = cos (x)

True

Solution steps

2 sin^{2} (\frac{x}{2}) + 4 cos^{2} (\frac{x}{2}) - 3 = cos (x)

Let:

u = \frac{x}{2}

2 sin^{2} (u) + 4 cos^{2} (u) - 3 = cos (2 u)

Prove

2 sin^{2} (u) + 4 cos^{2} (u) - 3 = cos (2 u) :

True

2 sin^{2} (u) + 4 cos^{2} (u) - 3 = cos (2 u)

Manipulating left side

2 sin^{2} (u) + 4 cos^{2} (u) - 3

Rewrite using trig identities

2 sin^{2} (u) + 4 cos^{2} (u) - 3

Use the Pythagorean identity:

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

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

= 2 sin^{2} (u) + 4 (1 - sin^{2} (u)) - 3

Simplify

2 sin^{2} (u) + 4 (1 - sin^{2} (u)) - 3 : - 2 sin^{2} (u) + 1

2 sin^{2} (u) + 4 (1 - sin^{2} (u)) - 3

Expand

4 (1 - sin^{2} (u)) : 4 - 4 sin^{2} (u)

4 (1 - sin^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (u)

= 4 \cdot 1 - 4 sin^{2} (u)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (u)

= 2 sin^{2} (u) + 4 - 4 sin^{2} (u) - 3

Simplify

2 sin^{2} (u) + 4 - 4 sin^{2} (u) - 3 : - 2 sin^{2} (u) + 1

2 sin^{2} (u) + 4 - 4 sin^{2} (u) - 3

Group like terms

= 2 sin^{2} (u) - 4 sin^{2} (u) + 4 - 3

Add similar elements:

2 sin^{2} (u) - 4 sin^{2} (u) = - 2 sin^{2} (u)

= - 2 sin^{2} (u) + 4 - 3

Add/Subtract the numbers:

4 - 3 = 1

= - 2 sin^{2} (u) + 1

= - 2 sin^{2} (u) + 1

= - 2 sin^{2} (u) + 1

Use the Double Angle identity:

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

= cos (2 u)

= cos (2 u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

2 sin^{2} (\frac{x}{2}) + 4 cos^{2} (\frac{x}{2}) - 3 = cos (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (2 u) = \frac{( 2 tan ( u ) )}{1 - tan ^{2} ( u )}

sin (4 x) = 2 sin (2 x)

\frac{1}{- sin ( x )} = sec (x) tan (x)

6 cos (θ) = 6 cos (- θ)

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

Is 2sin^2(x/2)+4cos^2(x/2)-3=cos(x) ?
The answer to whether 2sin^2(x/2)+4cos^2(x/2)-3=cos(x) is True