Verify that each equation is an identity.
(sin⁴ α - cos⁴...

Question

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Accepted Answer

Hello, everyone we are asked is the given equation and identity. We are given the fraction C can't raise to the fourth power X minus tangent raised to the fourth power X divided by the C can't squared X plus the tangent squared X equals one. Our answer choices are a yes B no. For this to be considered an identity, we need to make the left hand side equal the right hand side. I'm gonna start by working with the left hand side. So C can't raise to the fourth power X minus tangent raise the fourth power X divided by the C can't squared X plus tangent squared X. In our numerator, I have the difference of squares. I recall I can factor that into the product of a sum in a difference. So I will take the square root of the first term. So C can't raise to the fourth power X will become C can't squared X as the first term in both sets of parentheses. Then one set of parentheses gets a plus sign, one gets a minus sign. And then I take the square root of the second term, the square root of tangent to the fourth power X will be tangent squared X. And that's all still divided by C can't squared X plus tangent squared X. Now I notice a common factor I can reduce the C can squared X plus tangent squared X in the numerator with the one in the denominator. And when I do that, I'm left with the C can't squared X minus tangent squared X. And I recall an identity that stated that the C can't squared X minus the tangent squared of X equals one. It's a Pythagorean identity. So we can say that this equals one. And so now our left hand side will match our right hand side and this is answer choice. A yes, it is an identity. Have a nice day.

Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Key Concepts

Trigonometric Identities

Difference of Squares

Simplifying Rational Expressions

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