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\( \begin{array}{ll}\text { (b) } \frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \text { (An important result to remember) } \\ \text { (c) } \frac{1}{\sin ^{2} \theta}-\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=1 & \text { (d) } \frac{\tan x \cdot \cos x}{\sin x}=1\end{array} \)

Ask by Weber Thornton. in South Africa
Mar 15,2025

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All three equations are valid trigonometric identities: - (b) \(\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\) - (c) \(\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} = 1\) - (d) \(\frac{\tan x \cdot \cos x}{\sin x} = 1\)

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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!

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