Prove that: If sin^-1 x+sin^-1 y+sin^-1 z=(3 π)/(2), then prove that x y+y z+z x=3

Name: Problem 201, Chapter 2: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-08T18:05:21-08:00
Duration: 54 s
Channel: Rukhmani Jain
Description: Prove that: If sin^-1 x+sin^-1 y+sin^-1 z=(3 π)/(2), then prove that x y+y z+z x=3

step 1 We are given with the condition that is sine inverse x plus sine inverse y plus sine inverse z i

Prove that: If sin^-1 x+sin^-1 y+sin^-1 z=(3 π)/(2), then...

Key Concepts

Equality Conditions in Constrained Sums

When an equation involves the sum of functions with known maximum or minimum values, it is often useful to consider the conditions under which the extreme sum can be achieved. In this context, since sin?¹ x can at most be ?/2 for x within [-1, 1], a sum of three such terms equaling 3?/2 implies each term must be ?/2, mandating that each x, y, and z is 1. This reasoning is fundamental in proofs that involve equating a sum to the maximum possible value and extracting the necessary identities among the variables.

Monotonicity of the Arcsine Function

The arcsine function is strictly increasing on its domain. This property implies that larger input values produce larger output angles. Monotonicity is important when dealing with equations that sum several instances of arcsine, because it ensures that the maximum (or minimum) possible value of each term is attained at the upper (or lower) end of the domain. In problems where the sum reaches an extreme value, this property may force each individual term to be at its own extreme.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arcsine (sin?¹), are functions that 'undo' the action of the corresponding trigonometric function. They take a value from the range of the original trigonometric function and return an angle, typically within a restricted range. This concept is central to converting between the numerical and angular representations of trigonometric relationships and is frequently used in solving equations involving trigonometric expressions.

Domain and Range of the Arcsine Function

The arcsine function, sin?¹ x, is specifically defined for x values in the interval [-1, 1] and its range is restricted to [-?/2, ?/2]. This restriction is crucial for ensuring that the arcsine is a properly defined function (i.e., one-to-one) and plays a key role in understanding any constraints present in equations that involve sums of arcsine terms.

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