Is 2sin^2(θ)-1=1-2cos^2(θ) ?

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

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prove 2sin^2(θ)-1=1-2cos^2(θ)

prove

True

Solution steps

2 sin^{2} (θ) - 1 = 1 - 2 cos^{2} (θ)

Manipulating left side

2 sin^{2} (θ) - 1

Rewrite using trig identities

2 sin^{2} (θ) - 1

Use the Pythagorean identity:

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

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

= 2 (1 - cos^{2} (θ)) - 1

Simplify

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

2 (1 - cos^{2} (θ)) - 1

Expand

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

2 (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (θ)

= 2 \cdot 1 - 2 cos^{2} (θ)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (θ)

= 2 - 2 cos^{2} (θ) - 1

Simplify

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

2 - 2 cos^{2} (θ) - 1

Group like terms

= - 2 cos^{2} (θ) + 2 - 1

Add/Subtract the numbers:

2 - 1 = 1

= - 2 cos^{2} (θ) + 1

= - 2 cos^{2} (θ) + 1

= - 2 cos^{2} (θ) + 1

= - 2 cos^{2} (θ) + 1

= 1 - 2 cos^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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