State whether or not the equation is an identity. If it is an identity, prove it. (sinx+cosx)^2=

step 1 We're asked, does sign of x plus cosine of x all squared equal sine squared plus cosine squared

State whether or not the equation is an identity. If it is...

Key Concepts

Trigonometric Identity

A trigonometric identity is an equation involving trigonometric functions that holds true for every value of the variable within its domain. In analyzing whether an equation is an identity, one must verify that after appropriate algebraic manipulation and simplification, both sides of the equation are equivalent for all inputs. This concept is foundational in trigonometry for simplifying expressions and proving equations.

Pythagorean Identity

The Pythagorean identity, sin²x + cos²x = 1, is one of the most fundamental identities in trigonometry. It relates the square of the sine and cosine functions of any angle and is frequently used as a tool in verifying other trigonometric identities. Understanding and applying this identity is essential to simplify and manipulate trigonometric expressions.

Expansion of a Binomial

The expansion of a binomial, using the algebraic identity (a + b)² = a² + 2ab + b², is a critical technique in both algebra and trigonometry. When applied to trigonometric functions, it transforms expressions like (sin x + cos x)² into sin²x + 2 sin x cos x + cos²x; this reveals additional terms that must be handled when determining if an equation qualifies as an identity.

Conditional Equation vs. Identity

A conditional equation is true only for certain values of the variable, whereas an identity holds for all values in the domain. Recognizing the difference between these two is key when solving or verifying equations—in this case, after expansion, the extra term 2 sin x cos x indicates that the original equation is not an identity because it does not hold true for all values of x.

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