Sin Cos Identity D91690

1. **State the problem:** Prove or simplify the expression $$\sin^2 x - 3\cos^2 x$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$ This identity allows us to express one trigonometric function in terms of the other. 3. **Rewrite the expression using the identity:** Replace $$\sin^2 x$$ with $$1 - \cos^2 x$$: $$\sin^2 x - 3\cos^2 x = (1 - \cos^2 x) - 3\cos^2 x$$ 4. **Simplify the expression:** $$= 1 - \cos^2 x - 3\cos^2 x = 1 - 4\cos^2 x$$ 5. **Final simplified form:** $$\sin^2 x - 3\cos^2 x = 1 - 4\cos^2 x$$ This is the simplified form of the given expression using the Pythagorean identity. 6. **Alternative form:** Using $$\cos^2 x = 1 - \sin^2 x$$, we can also write: $$\sin^2 x - 3\cos^2 x = \sin^2 x - 3(1 - \sin^2 x) = \sin^2 x - 3 + 3\sin^2 x = 4\sin^2 x - 3$$ So the expression can also be written as $$4\sin^2 x - 3$$. This completes the proof and simplification.