# How to prove the identity? (sin(3x))/sin(x) +(cos(3x)/cos(x))=4(1-2sin^2(x)

Please see the proof below.

We need

Consequently,

Applying a common denominator

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To prove the identity (sin(3x))/sin(x) + (cos(3x)/cos(x)) = 4(1 - 2sin^2(x)), you can start by expressing sin(3x) and cos(3x) using trigonometric identities:

sin(3x) = 3sin(x) - 4sin^3(x) cos(3x) = 4cos^3(x) - 3cos(x)

Then, substitute these expressions into the left-hand side of the given identity:

(sin(3x))/sin(x) + (cos(3x)/cos(x)) = (3sin(x) - 4sin^3(x))/sin(x) + (4cos^3(x) - 3cos(x))/cos(x)

Now, simplify each term:

= 3 - 4sin^2(x) + 4cos^2(x) - 3sin^2(x) = 3 - 3sin^2(x) + 4cos^2(x) - 4sin^2(x)

Use the Pythagorean identity sin^2(x) + cos^2(x) = 1:

= 3 - 3(1 - cos^2(x)) + 4cos^2(x) - 4(1 - cos^2(x)) = 3 - 3 + 3cos^2(x) + 4cos^2(x) - 4 + 4cos^2(x)

Combine like terms:

= 7cos^2(x) - 4

Now, recall the Pythagorean identity again:

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

Substitute this into the expression:

= 7(1 - sin^2(x)) - 4 = 7 - 7sin^2(x) - 4

= 3 - 7sin^2(x)

Now, factor out 1 - 2sin^2(x):

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

This matches the right-hand side of the given identity, proving its validity.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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