Is 4cos^2(x)+cos(2x)=5-6sin^2(x) ?

The answer to whether 4cos^2(x)+cos(2x)=5-6sin^2(x) is True

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prove 4cos^2(x)+cos(2x)=5-6sin^2(x)

prove

4 cos^{2} (x) + cos (2 x) = 5 - 6 sin^{2} (x)

True

Solution steps

4 cos^{2} (x) + cos (2 x) = 5 - 6 sin^{2} (x)

Manipulating left side

4 cos^{2} (x) + cos (2 x)

Rewrite using trig identities

4 cos^{2} (x) + cos (2 x)

Use the Double Angle identity:

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

= 4 cos^{2} (x) + 2 cos^{2} (x) - 1

Add similar elements:

4 cos^{2} (x) + 2 cos^{2} (x) = 6 cos^{2} (x)

= 6 cos^{2} (x) - 1

= 6 cos^{2} (x) - 1

Manipulating right side

5 - 6 sin^{2} (x)

Rewrite using trig identities

5 - 6 sin^{2} (x)

Use the Pythagorean identity:

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

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

= 5 - 6 (1 - cos^{2} (x))

Simplify

5 - 6 (1 - cos^{2} (x)) : 6 cos^{2} (x) - 1

5 - 6 (1 - cos^{2} (x))

Expand

- 6 (1 - cos^{2} (x)) : - 6 + 6 cos^{2} (x)

- 6 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 6, b = 1, c = cos^{2} (x)

= - 6 \cdot 1 - (- 6) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 6 \cdot 1 + 6 cos^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

= - 6 + 6 cos^{2} (x)

= 5 - 6 + 6 cos^{2} (x)

Subtract the numbers:

5 - 6 = - 1

= 6 cos^{2} (x) - 1

= 6 cos^{2} (x) - 1

= 6 cos^{2} (x) - 1

We showed that the two sides could take the same form

\Rightarrow True

Is 4cos^2(x)+cos(2x)=5-6sin^2(x) ?
The answer to whether 4cos^2(x)+cos(2x)=5-6sin^2(x) is True