Is cos^2(B)-sin^2(B)=2cos^2(B)-1 ?

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

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prove cos^2(B)-sin^2(B)=2cos^2(B)-1

prove

True

Solution steps

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

Manipulating left side

cos^{2} (B) - sin^{2} (B)

Rewrite using trig identities

cos^{2} (B) - sin^{2} (B)

Use the Pythagorean identity:

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

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

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

Simplify

cos^{2} (B) - (1 - cos^{2} (B)) : 2 cos^{2} (B) - 1

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

cos^{2} (B) + cos^{2} (B) = 2 cos^{2} (B)

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Is cos^2(B)-sin^2(B)=2cos^2(B)-1 ?
The answer to whether cos^2(B)-sin^2(B)=2cos^2(B)-1 is True