Is sin^2(x)-4cos^2(x)=5sin^2(x)-4 ?

The answer to whether sin^2(x)-4cos^2(x)=5sin^2(x)-4 is True

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prove sin^2(x)-4cos^2(x)=5sin^2(x)-4

prove

sin^{2} (x) - 4 cos^{2} (x) = 5 sin^{2} (x) - 4

True

Solution steps

sin^{2} (x) - 4 cos^{2} (x) = 5 sin^{2} (x) - 4

Manipulating left side

sin^{2} (x) - 4 cos^{2} (x)

Rewrite using trig identities

sin^{2} (x) - 4 cos^{2} (x)

Use the Pythagorean identity:

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

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

= sin^{2} (x) - 4 (1 - sin^{2} (x))

Simplify

sin^{2} (x) - 4 (1 - sin^{2} (x)) : 5 sin^{2} (x) - 4

sin^{2} (x) - 4 (1 - sin^{2} (x))

Expand

- 4 (1 - sin^{2} (x)) : - 4 + 4 sin^{2} (x)

- 4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = 1, c = sin^{2} (x)

= - 4 \cdot 1 - (- 4) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 sin^{2} (x)

= sin^{2} (x) - 4 + 4 sin^{2} (x)

Simplify

sin^{2} (x) - 4 + 4 sin^{2} (x) : 5 sin^{2} (x) - 4

sin^{2} (x) - 4 + 4 sin^{2} (x)

Group like terms

= sin^{2} (x) + 4 sin^{2} (x) - 4

Add similar elements:

sin^{2} (x) + 4 sin^{2} (x) = 5 sin^{2} (x)

= 5 sin^{2} (x) - 4

= 5 sin^{2} (x) - 4

= 5 sin^{2} (x) - 4

= 5 sin^{2} (x) - 4

We showed that the two sides could take the same form

\Rightarrow True

Is sin^2(x)-4cos^2(x)=5sin^2(x)-4 ?
The answer to whether sin^2(x)-4cos^2(x)=5sin^2(x)-4 is True