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\( \begin{array}{ll}\text { (e) } \sin ^{4} x-\cos ^{4} x=\sin ^{2} x-\cos ^{2} x & \text { (f) } \sin ^{2} \theta+\sin ^{2} \theta \cdot \tan ^{2} \theta=\tan ^{2} \theta \\ \text { (g) } \cos \theta(1+\tan \theta)=\cos \theta+\sin \theta & \text { (h) } \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}=\frac{1-\cos \theta}{1+\cos \theta} \\ \text { (i) } \frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}=\frac{2}{\sin ^{2} \theta} & \text { (j) } \frac{1}{\sin \theta}+\frac{1}{\tan \theta}=\frac{1+\cos \theta}{\sin \left(180^{\circ}-\theta\right)}\end{array} \)

Ask by Moran Lyons. in South Africa
Mar 15,2025

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All the given trigonometric identities are verified as true through algebraic manipulation and the use of fundamental trigonometric identities.

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Did you know that trigonometric identities have been studied for centuries? Ancient Greek mathematicians like Hipparchus explored these relationships, laying the groundwork for modern trigonometry. By the Middle Ages, scholars in the Islamic Golden Age expanded on this knowledge, developing sophisticated trigonometric tables and concepts that we still use today. So, next time you're solving a trigonometric equation, give a nod to these ancient thinkers! If you're diving into these identities, remember to keep an eye out for common mistakes! One of the biggest pitfalls is forgetting to apply the Pythagorean identity, which states that \(\sin^2 \theta + \cos^2 \theta = 1\). This foundational identity is like your trusty toolkit—use it wisely to simplify expressions and avoid missteps. Happy calculating!

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