\( \begin{array}{ll}\text { (b) } \frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \text { (An important result to remember) } \\ \text { (c) } \frac{1}{\sin ^{2} \theta}-\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=1 & \text { (d) } \frac{\tan x \cdot \cos x}{\sin x}=1\end{array} \)
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The Deep Dive
Did you know that trigonometric identities have been used since ancient times by civilizations like the Greeks and Indians to solve problems related to astronomy and engineering? These identities were crucial for navigation and understanding the cosmos, laying the groundwork for modern mathematics! When applying trigonometric identities, a common mistake is to overlook the fundamental Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This mistake can lead to incorrect conclusions. Always double-check your derivations and remember to simplify expressions fully to avoid these pitfalls!