Is sin^2(x)-2cos^2(x)=3sin^2(x)-2 ?

The answer to whether sin^2(x)-2cos^2(x)=3sin^2(x)-2 is True

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prove sin^2(x)-2cos^2(x)=3sin^2(x)-2

prove

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

True

Solution steps

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

sin^{2} (x) - 2 (1 - sin^{2} (x)) : 3 sin^{2} (x) - 2

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

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)

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

sin^{2} (x) - 2 + 2 sin^{2} (x) : 3 sin^{2} (x) - 2

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

Group like terms

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

Add similar elements:

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

We showed that the two sides could take the same form

\Rightarrow True

Is sin^2(x)-2cos^2(x)=3sin^2(x)-2 ?
The answer to whether sin^2(x)-2cos^2(x)=3sin^2(x)-2 is True