Is 7cos^2(x)+5sin^2(x)=6+cos(2x) ?

The answer to whether 7cos^2(x)+5sin^2(x)=6+cos(2x) is True

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prove 7cos^2(x)+5sin^2(x)=6+cos(2x)

prove

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

True

Solution steps

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

Manipulating left side

7 cos^{2} (x) + 5 sin^{2} (x)

Rewrite using trig identities

7 cos^{2} (x) + 5 sin^{2} (x)

Use the Pythagorean identity:

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

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

= 7 (1 - sin^{2} (x)) + 5 sin^{2} (x)

Simplify

7 (1 - sin^{2} (x)) + 5 sin^{2} (x) : 7 - 2 sin^{2} (x)

7 (1 - sin^{2} (x)) + 5 sin^{2} (x)

Expand

7 (1 - sin^{2} (x)) : 7 - 7 sin^{2} (x)

7 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 7, b = 1, c = sin^{2} (x)

= 7 \cdot 1 - 7 sin^{2} (x)

Multiply the numbers:

7 \cdot 1 = 7

= 7 - 7 sin^{2} (x)

= 7 - 7 sin^{2} (x) + 5 sin^{2} (x)

Add similar elements:

- 7 sin^{2} (x) + 5 sin^{2} (x) = - 2 sin^{2} (x)

= 7 - 2 sin^{2} (x)

= 7 - 2 sin^{2} (x)

= 7 - 2 sin^{2} (x)

Manipulating right side

6 + cos (2 x)

Rewrite using trig identities

6 + cos (2 x)

Use the Double Angle identity:

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

= 6 + 1 - 2 sin^{2} (x)

Simplify

= - 2 sin^{2} (x) + 7

= - 2 sin^{2} (x) + 7

We showed that the two sides could take the same form

\Rightarrow True

Is 7cos^2(x)+5sin^2(x)=6+cos(2x) ?
The answer to whether 7cos^2(x)+5sin^2(x)=6+cos(2x) is True