Is 5cos^2(x)-5sin^2(x)=10cos^2(x)-5 ?

The answer to whether 5cos^2(x)-5sin^2(x)=10cos^2(x)-5 is True

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prove 5cos^2(x)-5sin^2(x)=10cos^2(x)-5

prove

5 cos^{2} (x) - 5 sin^{2} (x) = 10 cos^{2} (x) - 5

True

Solution steps

5 cos^{2} (x) - 5 sin^{2} (x) = 10 cos^{2} (x) - 5

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= 5 cos^{2} (x) - 5 (1 - cos^{2} (x))

Simplify

5 cos^{2} (x) - 5 (1 - cos^{2} (x)) : 10 cos^{2} (x) - 5

5 cos^{2} (x) - 5 (1 - cos^{2} (x))

Expand

- 5 (1 - cos^{2} (x)) : - 5 + 5 cos^{2} (x)

- 5 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 5, b = 1, c = cos^{2} (x)

= - 5 \cdot 1 - (- 5) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 5 \cdot 1 + 5 cos^{2} (x)

Multiply the numbers:

5 \cdot 1 = 5

= - 5 + 5 cos^{2} (x)

= 5 cos^{2} (x) - 5 + 5 cos^{2} (x)

Simplify

5 cos^{2} (x) - 5 + 5 cos^{2} (x) : 10 cos^{2} (x) - 5

5 cos^{2} (x) - 5 + 5 cos^{2} (x)

Group like terms

= 5 cos^{2} (x) + 5 cos^{2} (x) - 5

Add similar elements:

5 cos^{2} (x) + 5 cos^{2} (x) = 10 cos^{2} (x)

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

We showed that the two sides could take the same form

\Rightarrow True

Is 5cos^2(x)-5sin^2(x)=10cos^2(x)-5 ?
The answer to whether 5cos^2(x)-5sin^2(x)=10cos^2(x)-5 is True