1. **Problem statement:** Prove the trigonometric identity: $$\tan\theta - \tan(90^\circ - \theta) = \frac{\sin\theta + \cos\theta}{\cos\theta \cos(90^\circ - \theta)} = 1$$ 2. **Recall formulas and identities:** - $\tan\theta = \frac{\sin\theta}{\cos\theta}$ - $\tan(90^\circ - \theta) = \cot\theta = \frac{\cos\theta}{\sin\theta}$ - $\cos(90^\circ - \theta) = \sin\theta$ 3. **Rewrite the left side:** $$\tan\theta - \tan(90^\circ - \theta) = \frac{\sin\theta}{\cos\theta} - \frac{\cos\theta}{\sin\theta}$$ 4. **Find common denominator and combine:** $$= \frac{\sin^2\theta - \cos^2\theta}{\cos\theta \sin\theta}$$ 5. **Use identity for numerator:** $$\sin^2\theta - \cos^2\theta = -(\cos^2\theta - \sin^2\theta) = -\cos 2\theta$$ So, $$\tan\theta - \tan(90^\circ - \theta) = \frac{-\cos 2\theta}{\cos\theta \sin\theta}$$ 6. **Rewrite denominator using double angle:** Recall $\sin 2\theta = 2 \sin\theta \cos\theta$, so $$\cos\theta \sin\theta = \frac{\sin 2\theta}{2}$$ 7. **Substitute back:** $$= \frac{-\cos 2\theta}{\frac{\sin 2\theta}{2}} = -\frac{2 \cos 2\theta}{\sin 2\theta}$$ 8. **Check the right side given:** The problem states the expression equals 1, but from above we have $$\tan\theta - \tan(90^\circ - \theta) = -\frac{2 \cos 2\theta}{\sin 2\theta}$$ This is a simplified form, not equal to 1 generally. 9. **Check the middle expression:** $$\frac{\sin\theta + \cos\theta}{\cos\theta \cos(90^\circ - \theta)} = \frac{\sin\theta + \cos\theta}{\cos\theta \sin\theta}$$ This is different from the left side expression. 10. **Conclusion:** The given expressions are not equal as stated. The left side simplifies to $$\frac{\sin^2\theta - \cos^2\theta}{\cos\theta \sin\theta} = -\frac{\cos 2\theta}{\cos\theta \sin\theta}$$ The middle expression is $$\frac{\sin\theta + \cos\theta}{\cos\theta \sin\theta}$$ They are not equal in general. **Final answer:** The identity as stated is not true for all $\theta$. --- **Slug:** trigonometric-identity **Subject:** trigonometry "desmos": {"latex": "y=\tan\theta - \tan(90^\circ - \theta)", "features": {"intercepts": true, "extrema": true}}, "q_count": 1