Is 2cos^2(x/2)=1+cos(x) ?

The answer to whether 2cos^2(x/2)=1+cos(x) is True

Popular >

prove 2cos^2(x/2)=1+cos(x)

prove

2 cos^{2} (\frac{x}{2}) = 1 + cos (x)

True

Solution steps

2 cos^{2} (\frac{x}{2}) = 1 + cos (x)

Let:

u = \frac{x}{2}

2 cos^{2} (u) = 1 + cos (2 u)

Prove

2 cos^{2} (u) = 1 + cos (2 u) :

True

2 cos^{2} (u) = 1 + cos (2 u)

Manipulating right side

1 + cos (2 u)

Rewrite using trig identities

1 + cos (2 u)

Use the Double Angle identity:

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

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

Simplify

1 + 2 cos^{2} (u) - 1 : 2 cos^{2} (u)

1 + 2 cos^{2} (u) - 1

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (u)

= 2 cos^{2} (u)

= 2 cos^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

2 cos^{2} (\frac{x}{2}) = 1 + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (a) + \frac{cos ( a )}{1 + sin ( a )} = sec (a)

\frac{cos ^{2} ( A )}{cos ( A ) - sin ( A )} + \frac{sin ( A )}{1 - cot ( A )} = sin (A) + cos (A)

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

cos (x - \frac{π}{6}) - sin (x + \frac{π}{3}) = 0

cot^{2} (α) + 1 = \frac{1}{sin ^{2} ( α )}