Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on the following concepts:

• Addition and Subtraction Theorems: Applying formulas for ${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$, and ${\cot }^{-1}$ to combine multiple terms.
• Reciprocal Relationships: Using the relationships between ${\csc }^{-1}$, ${\sec }^{-1}$, ${\cot }^{-1}$ and their primary counterparts (${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$).
• Identity Verification: Proving that the sum or difference of specific inverse trigonometric values equals a constant (e.g., $\pi /2$, $\pi /4$).
• Negative Arguments: Handling properties like ${\sin }^{-1}(-x)=-{\sin }^{-1}(x)$ and ${\cos }^{-1}(-x)=\pi -{\cos }^{-1}(x)$.

• Algebraic Substitution: Evaluating complex inverse trigonometric expressions by substituting specific values for $x$.

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Important Formulas

Below are the key formulas used in this exercise:

| Cosine | ${\cos }^{-1}x\pm {\cos }^{-1}y={\cos }^{-1}\left(xy\mp \sqrt{(1-x^2)(1-y^2)}\right)$ |

| Tangent | ${\tan }^{-1}x\pm {\tan }^{-1}y={\tan }^{-1}\left(\frac{x\pm y}{1\mp xy}\right)$ |

| Cotangent | ${\cot }^{-1}x+{\cot }^{-1}y={\cot }^{-1}\left(\frac{xy-1}{x+y}\right)$ | | Secant | ${\sec }^{-1}x+{\sec }^{-1}y={\cos }^{-1}\left(\frac{1-\sqrt{(x^2-1)(y^2-1)}}{xy}\right)$ | | Cosecant | ${\mathrm{cosec}}^{-1}x+{\mathrm{cosec}}^{-1}y={\sin }^{-1}\left[\frac{\sqrt{y^2-1}+\sqrt{x^2-1}}{xy}\right]$ |

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Summary

This exercise provides practice in manipulating inverse trigonometric functions through their respective addition and subtraction laws. The primary strategy involves identifying the appropriate formula based on the function type and simplifying the resulting algebraic expression. For functions like ${\sec }^{-1}$ and ${\csc }^{-1}$, it is often helpful to convert them into ${\cos }^{-1}$ and ${\sin }^{-1}$ forms.

A common pattern in these problems is the verification of the identity ${\sin }^{-1}x+{\cos }^{-1}x=\pi /2$ or similar results involving complementary angles. When solving equations, ensure that the final result is checked against the principal value branches of the inverse functions.