Exploring Trigonometric Identities

Trigonometric identities often present fascinating challenges that require a blend of algebraic manipulation and understanding of trigonometric properties. They allow us to transform and simplify expressions, making it easier to solve equations and understand the relationships between different trigonometric functions. 

 If `sin(x)+cos(x)=\sqrt{3}-1`, then what is `sin(2x)`?

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Answer: `3-2\sqrt{3}`

Explanation: 

`(sin(x)+cos(x))^2=(\sqrt{3}-1)^2`

`\underbrace{\sin^2(x)+\cos^2(x)}_1+\underbrace{2\sin(x)\cos(x)}_\sin(2x)=3-2\sqrt{3}+1`

`1+\sin(2x)=3-2\sqrt{3}+1`

`\sin(2x)=3-2\sqrt{3}`