Prove the identity. (sin x+cos x)^(2)=1+sin 2x

Expand using binomial formula: We start by expanding the left side of the equation using the binomial formula (a + b)^2 = a^2 + 2ab + b^2. (sin x + cos x)^2 = (sin x)^2 + 2(sin x)(cos x) + (cos x)^2Apply Pythagorean identity: Next, we recognize that (sin x)^2 and (cos x)^2 are part of the Pythagorean identity, which states that (sin x)^2 + (cos x)^2 = 1. (sin x)^2 + (cos x)^2 = 1Use double angle formula: We also know that 2(sin x)(cos x) is the double angle formula for sine, which states that sin 2x = 2(sin x)(cos x). 2(sin x)(cos x) = sin 2xSubstitute into expanded equation: Now, we substitute the Pythagorean identity and the double angle formula into our expanded equation. (sin x + cos x)^2 = 1 + sin 2xProve the identity: We have shown that the left side of the equation, when expanded and simplified using known trigonometric identities, equals the right side of the equation. Therefore, the identity (sin x + cos x)^2 = 1 + sin 2x is proven.

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Q. Prove the identity.

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(\sin x+\cos x)^{2}=1+\sin 2x

Expand using binomial formula: We start by expanding the left side of the equation using the binomial formula $(a + b)^2 = a^2 + 2ab + b^2$ . $\newline$ (\sin x + \cos x)^\(2 = (\sin x)^ $2$ + $2$ (\sin x)(\cos x) + (\cos x)^ $2$
Apply Pythagorean identity: Next, we recognize that $(\sin x)^2$ and $(\cos x)^2$ are part of the Pythagorean identity, which states that $(\sin x)^2 + (\cos x)^2 = 1$ . $\newline$ $(\sin x)^2 + (\cos x)^2 = 1$
Use double angle formula: We also know that $2(\sin x)(\cos x)$ is the double angle formula for sine, which states that $\sin 2x = 2(\sin x)(\cos x)$ . $\newline$ $2(\sin x)(\cos x) = \sin 2x$
Substitute into expanded equation: Now, we substitute the Pythagorean identity and the double angle formula into our expanded equation. $\newline$ $(\sin x + \cos x)^2 = 1 + \sin 2x$
Prove the identity: We have shown that the left side of the equation, when expanded and simplified using known trigonometric identities, equals the right side of the equation. $\newline$ Therefore, the identity $(\sin x + \cos x)^2 = 1 + \sin 2x$ is proven.