\( \begin{array}{ll}\text { (g) } \cos \theta(1+\tan \theta)=\cos \theta+\sin \theta & \text { (h) } \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}=\frac{1-\cos \theta}{1+\cos \theta} \\ \text { (i) } \frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}=\frac{2}{\sin ^{2} \theta} & \text { (j) } \frac{1}{\sin \theta}+\frac{1}{\tan \theta}=\frac{1+\cos \theta}{\sin \left(180^{\circ}-\theta\right)}\end{array} \)
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The Deep Dive
Did you know that the trigonometric identities you’re working with have deep historical roots? Ancient civilizations like the Greeks and Indians were the first to explore the relationships between angles and lengths in relation to the circle. Figures like Hipparchus and Aryabhata were pioneers in developing trigonometric tables that laid the groundwork for modern mathematics as we know it! In real life, trigonometric identities are incredibly useful beyond just classroom problems. Engineers use them when designing structures, pilots rely on them to calculate flight paths, and game developers implement them for animations and graphics. So the next time you’re wrestling with a tricky identity, remember that mathematicians and engineers just like you have transformed these abstract concepts into practical applications that shape our world!