Question

Complete the proof of the identity by choosing the Rule that justifies each step. Sin^2x + 4cos^2x = 4 -3sin^2x To see a detailed description of a Rule in the Rule menu, select the corresponding question mark. Statement Rule sin^2 x + 4cos^2 x sin^2 x + 4 (1 -sin^2 x) Rule? sin^2 x + 4 -4sin^2 x Rule? 4 -3sin^2 x Rule? Reciprocal identities: sin u = 1/csc u cos u = 1/sec u tan u = 1/cot u csc u = 1/sin u sec u = 1/cos u cot u = 1/tan u Quotient identities: tan u = sin u/cos u cot u = cos u/sin u Pythagorean identities: sin^2u + cos^2u = 1 tan^2u + 1 = sec^2 u cot^2 u + 1 = csc^2 u Odd/Even function identities: sin(-u) = -sin(u) cos(-u) = cos(u) tan(-u) = -tan(u) csc(-u) = -csc(u) sec(-u) = sec(u) cot(-u) = -cot(u)

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Answer 1

The correct proof of the identity is as follows:

Sin^2x + 4cos^2x = 4 -3sin^2x
sin^2 x + 4 (1 -sin^2 x) (Quotient identity: sin^2 x = 1 - sin^2 x)
sin^2 x + 4 -4sin^2 x (Addition Property of Equality)
4 -3sin^2 x (Substitution)

Each step in this proof is justified by the following rules:

The first step uses the Quotient identity sin^2 x = 1 - sin^2 x, which states that the sine squared of an angle is equal to 1 minus the sine squared of the angle.
The second step uses the Addition Property of Equality, which states that if two expressions are equal to a third expression, then their sum is also equal to that third expression.
The third step uses substitution, where the value of an expression is replaced by its equivalent value.
Therefore, the proof is complete and the identity is proven to be true.