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\[ \begin{array}{rlr} \text { LHS } & =\frac{1+\sin x}{\sin } \times \frac{1-\cos x}{1-\cos x} & \text { Notice that you are multiplying by } \\ & =\frac{1-\cos ^{2} x}{\sin x(1-\cos x)} & \text { Notice the difference of } 2 \text { squares. } \\ & =\frac{\sin ^{2} x}{\sin x(1-\cos x)} & \text { Notice the identity } 1-\cos ^{2} x=\sin \\ & =\frac{\sin x}{1-\cos x}=\text { RHS } & \end{array} \] EXERCISE 6 rove the following identities. \[ \begin{array}{l} \frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x} \\ \frac{\cos ^{2} x-\cos x-\sin ^{2} x}{2 \sin x \cdot \cos x+\sin x}=\frac{1}{\tan x}-\frac{1}{\sin x} \\ \sin x-\frac{1}{\sin x+\cos x}+\cos x=\frac{2 \sin x \cdot \cos x}{\sin x+\cos x} \\ \frac{\sin ^{2} x+\sin x-\cos ^{2} x}{2 \sin x \cdot \cos x-\cos x}=\frac{\sin x+1}{\cos x} \end{array} \]

Ask by Nguyen Harper. in South Africa
Mar 13,2025

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All four trigonometric identities have been proven step by step as follows: 1. **Identity 1:** \[ \frac{\cos x}{1 - \sin x} = \frac{1 + \sin x}{\cos x} \] - Cross-multiplied and used the Pythagorean identity to confirm equality. 2. **Identity 2:** \[ \frac{\cos^2 x - \cos x - \sin^2 x}{2 \sin x \cdot \cos x + \sin x} = \frac{1}{\tan x} - \frac{1}{\sin x} \] - Simplified both sides and showed they are equal. 3. **Identity 3:** \[ \sin x - \frac{1}{\sin x + \cos x} + \cos x = \frac{2 \sin x \cdot \cos x}{\sin x + \cos x} \] - Combined terms and used the Pythagorean identity to prove equality. 4. **Identity 4:** \[ \frac{\sin^2 x + \sin x - \cos^2 x}{2 \sin x \cdot \cos x - \cos x} = \frac{\sin x + 1}{\cos x} \] - Factored and simplified both sides to confirm equality. All identities have been proven correctly.

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Bonus Knowledge

It’s always intriguing to dive into the world of trigonometric identities! These identities not only help streamline complex calculations but also reveal profound relationships between different functions. Ancient civilizations, like the Greeks, utilized these relationships in astronomy and engineering, laying a foundation for modern mathematics. The beauty of these identities lies in their applicability across various fields, whether it's in wave functions in physics or computer graphics where angles define shapes! When tackling these identities, a common pitfall is overlooking fundamental trigonometric identities, like Pythagorean identities or the reciprocal identities (e.g., sin, cos, tan). Double-checking each step and simplifying smartly can save you from potential errors. Also, ensure to manipulate both sides of the equation in a balanced way, as it’s easy to inadvertently alter one side more than the other! Happy simplifying!

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