Is 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2 ?

The answer to whether 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2 is True

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prove 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2

prove

5 cos^{2} (θ) + 2 sin^{2} (θ) = 3 cos^{2} (θ) + 2

True

Solution steps

5 cos^{2} (θ) + 2 sin^{2} (θ) = 3 cos^{2} (θ) + 2

Manipulating left side

5 cos^{2} (θ) + 2 sin^{2} (θ)

Rewrite using trig identities

5 cos^{2} (θ) + 2 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= 5 cos^{2} (θ) + 2 (1 - cos^{2} (θ))

Simplify

5 cos^{2} (θ) + 2 (1 - cos^{2} (θ)) : 3 cos^{2} (θ) + 2

5 cos^{2} (θ) + 2 (1 - cos^{2} (θ))

Expand

2 (1 - cos^{2} (θ)) : 2 - 2 cos^{2} (θ)

2 (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (θ)

= 2 \cdot 1 - 2 cos^{2} (θ)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (θ)

= 5 cos^{2} (θ) + 2 - 2 cos^{2} (θ)

Simplify

5 cos^{2} (θ) + 2 - 2 cos^{2} (θ) : 3 cos^{2} (θ) + 2

5 cos^{2} (θ) + 2 - 2 cos^{2} (θ)

Group like terms

= 5 cos^{2} (θ) - 2 cos^{2} (θ) + 2

Add similar elements:

5 cos^{2} (θ) - 2 cos^{2} (θ) = 3 cos^{2} (θ)

= 3 cos^{2} (θ) + 2

= 3 cos^{2} (θ) + 2

= 3 cos^{2} (θ) + 2

= 3 cos^{2} (θ) + 2

We showed that the two sides could take the same form

\Rightarrow True

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Is 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2 ?
The answer to whether 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2 is True