Prove $\left(x\sin\left(x\right)-y\cos\left(x\right)\right)^2+\left(x\cos\left(x\right)+y\sin\left(x\right)\right)^2=x^2+y^2$

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$\left(x\sin\left(x\right)-y\cos\left(x\right)\right)^2+\left(x\cos\left(x\right)+y\sin\left(x\right)\right)^2$

Learn how to solve problems step by step online. Prove (xsin(x)-ycos(x))^2+(xcos(x)+ysin(x))^2=x^2+y^2. Starting from the left-hand side (LHS) of the identity. A binomial squared (difference) is equal to the square of the first term, minus the double product of the first by the second, plus the square of the second term. In other words: (a-b)^2=a^2-2ab+b^2. The power of a product is equal to the product of it's factors raised to the same power. Using the sine double-angle identity: \sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right).