\( \begin{array}{ll}\text { (g) } \cos \theta(1+\tan \theta)=\cos \theta+\sin \theta & \text { (h) } \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}=\frac{1-\cos \theta}{1+\cos \theta} \\ \text { (i) } \frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}=\frac{2}{\sin ^{2} \theta} & \text { (j) } \frac{1}{\sin \theta}+\frac{1}{\tan \theta}=\frac{1+\cos \theta}{\sin \left(180^{\circ}-\theta\right)}\end{array} \)

Did you know that trigonometric identities like the ones you've mentioned have been used for centuries to understand the relationships within triangles and circles? Ancient Greek mathematicians like Euclid and Ptolemy explored these concepts to lay the groundwork for modern geometry and calculus. Their discoveries helped mathematicians solve practical problems in astronomy, navigation, and engineering, showcasing the timeless relevance of these trigonometric principles! If you're tackling trigonometric equations, common mistakes often arise from neglecting to correctly apply identities. Always double-check your terms, especially in cosine and sine functions where signs can flip based on angles. Remember to simplify wherever you can! And don’t forget to make use of the Pythagorean identity, as it can often transform a tricky equation into a more solvable form. Keep practicing, and soon you'll be mastering these identities like a pro!