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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Manipulating left side

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

Rewrite using trig identities

cos (2 u) + 2 sin^{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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

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

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