Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

검증된 단계별 안내

Start by expanding the left side of the equation: \((1 + \sin x + \cos x)^2\). Use the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\).

Calculate each term: \(a^2 = 1^2 = 1\), \(b^2 = (\sin x)^2 = \sin^2 x\), \(c^2 = (\cos x)^2 = \cos^2 x\), \(2ab = 2 \cdot 1 \cdot \sin x = 2\sin x\), \(2ac = 2 \cdot 1 \cdot \cos x = 2\cos x\), \(2bc = 2 \cdot \sin x \cdot \cos x = 2\sin x \cos x\).

Combine these terms to get: \(1 + \sin^2 x + \cos^2 x + 2\sin x + 2\cos x + 2\sin x \cos x\).

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to simplify: \(1 + 1 + 2\sin x + 2\cos x + 2\sin x \cos x = 2 + 2\sin x + 2\cos x + 2\sin x \cos x\).

Now expand the right side: \(2(1 + \sin x)(1 + \cos x) = 2((1 + \sin x) + (1 + \sin x)\cos x) = 2(1 + \sin x + \cos x + \sin x \cos x)\). Simplify to see if both sides are equal.

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable within a certain domain. Common identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations.

Algebraic Expansion

Algebraic expansion involves applying the distributive property to multiply expressions, such as binomials. In the context of the given equation, expanding (1 + sin x + cos x)² requires using the formula (a + b)² = a² + 2ab + b², which helps in simplifying the left-hand side of the equation for comparison with the right-hand side.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied to obtain the original expression. In the context of verifying identities, recognizing common factors on both sides of the equation can simplify the verification process and help establish equality between the two sides.

Verify that each equation is an identity.(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)