Is 2cos^2(x)-cos(2x)=1 ?

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

prove 2cos^2(x)-cos(2x)=1

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Add similar elements:

2 cos^{2} (x) - 2 cos^{2} (x) = 0

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

p ro v e - cot^{2} (x) = 1 - csc^{2} (x)

p ro v e \frac{1}{tan ( 2 θ )} = \frac{cot ^{2} ( θ ) - 1}{2 cot ( θ )}

p ro v e (1 - cos (θ)) + sin^{2} (θ) = 2

p ro v e \frac{1}{sec ( θ )} = (sec (θ))^{- 1}

p ro v e sin (2 θ) + cos (2 θ) = 1