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

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

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

prove

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

True

Solution steps

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

Manipulating left side

2 cos^{2} (x) - 1

Rewrite using trig identities

2 cos^{2} (x) - 1

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

= 2 \cdot 1 - 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

Add/Subtract the numbers:

2 - 1 = 1

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos (x) (tan (x) sec (- x)) = sin (x) - 1

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

(1 + tan^{2} (x)) cot (x) = sec (x) csc (x)

cos (x - y) - cos (x + y) = 2 sin (x) sin (y)

tan (θ) csc (θ) = sec (θ)

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