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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Manipulating left side

2 sin^{2} (u) + cos (2 u)

Rewrite using trig identities

2 sin^{2} (u) + cos (2 u)

Use the Double Angle identity:

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

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

= cos (2 u) - (cos (2 u) - 1)

cos (2 u) - (cos (2 u) - 1) = 1

cos (2 u) - (cos (2 u) - 1)

- (cos (2 u) - 1) : - cos (2 u) + 1

- (cos (2 u) - 1)

Distribute parentheses

= - (cos (2 u)) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - cos (2 u) + 1

= cos (2 u) - cos (2 u) + 1

Add similar elements:

cos (2 u) - cos (2 u) = 0

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

sec (x) - cos (x) = sin (x) (tan (x))

sin (x) sin (3 x) = 4 sin (x) cos^{2} (x)

sec (\frac{π}{2} - t) = csc (t)

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

csc (x) sec (x) = tan (x) + cot (x)