Prove Trig Relation 1Fcdaa

1. **State the problem:** Prove the relation $$\tan^2(x) + \cos^2(x) + 2 = \frac{1}{\sin^2(x) \cos^2(x)}$$. 2. **Recall definitions and identities:** - $$\tan(x) = \frac{\sin(x)}{\cos(x)}$$ - Pythagorean identity: $$\sin^2(x) + \cos^2(x) = 1$$ 3. **Rewrite the left side using $$\tan^2(x)$$:** $$\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$$ So the left side is: $$\frac{\sin^2(x)}{\cos^2(x)} + \cos^2(x) + 2$$ 4. **Find a common denominator for the first two terms:** $$\frac{\sin^2(x)}{\cos^2(x)} + \cos^2(x) = \frac{\sin^2(x)}{\cos^2(x)} + \frac{\cos^4(x)}{\cos^2(x)} = \frac{\sin^2(x) + \cos^4(x)}{\cos^2(x)}$$ 5. **Add 2 to the expression:** $$\frac{\sin^2(x) + \cos^4(x)}{\cos^2(x)} + 2 = \frac{\sin^2(x) + \cos^4(x)}{\cos^2(x)} + \frac{2 \cos^2(x)}{\cos^2(x)} = \frac{\sin^2(x) + \cos^4(x) + 2 \cos^2(x)}{\cos^2(x)}$$ 6. **Simplify the numerator:** Group terms: $$\sin^2(x) + \cos^4(x) + 2 \cos^2(x) = \sin^2(x) + (\cos^2(x))^2 + 2 \cos^2(x)$$ Let $$y = \cos^2(x)$$, then numerator becomes: $$\sin^2(x) + y^2 + 2y$$ Using $$\sin^2(x) = 1 - y$$ from Pythagorean identity: $$1 - y + y^2 + 2y = 1 + y^2 + y$$ 7. **Rewrite numerator:** $$1 + y^2 + y = 1 + y + y^2$$ 8. **Recall the right side:** $$\frac{1}{\sin^2(x) \cos^2(x)} = \frac{1}{(1 - y) y}$$ 9. **Check if left side equals right side:** Left side is: $$\frac{1 + y + y^2}{y}$$ Right side is: $$\frac{1}{y(1 - y)}$$ 10. **Cross-multiply to verify equality:** $$ (1 + y + y^2)(1 - y) \stackrel{?}{=} 1 $$ 11. **Expand left side:** $$ (1)(1 - y) + y(1 - y) + y^2(1 - y) = (1 - y) + (y - y^2) + (y^2 - y^3) $$ Simplify: $$ 1 - y + y - y^2 + y^2 - y^3 = 1 - y^3 $$ 12. **Set equal to 1:** $$ 1 - y^3 = 1 \implies y^3 = 0 \implies y = 0 $$ 13. **Interpretation:** Since $$y = \cos^2(x)$$, the equality holds only if $$\cos^2(x) = 0$$, which is not generally true. 14. **Conclusion:** The given relation is not an identity for all $$x$$. It holds only when $$\cos^2(x) = 0$$, i.e., $$x = \frac{\pi}{2} + k\pi$$ for integers $$k$$. **Final answer:** The relation $$\tan^2(x) + \cos^2(x) + 2 = \frac{1}{\sin^2(x) \cos^2(x)}$$ is not true for all $$x$$, only for specific values where $$\cos^2(x) = 0$$.