Demystifying Inverse Trigonometric Functions: Solving cos?1x tan?1x

Inverse trigonometric functions play a crucial role in solving a wide range of mathematical problems. One such interesting equation is ( cos^{-1}x tan^{-1}x ). Let's dive into the solution of this intriguing equation and explore the fundamental concepts involved.

Understanding the Inverse Tangent Function

The inverse tangent function, often denoted as ( tan^{-1}x ), is used to find the angle in radians from a given tangent value. If we let ( tan^{-1}x t ), then it implies that ( x tan t ). From this, we can derive that:

[ frac{x}{sqrt{1 x^2}} cos t ]

Here, we can use the trigonometric identity of the secant in terms of tangent to further explore the relationship.

Solving ( cos^{-1}x tan^{-1}x )

Given the equation ( cos^{-1}x tan^{-1}x ), let's proceed with the solution step by step.

Let ( y cos^{-1}x ) which implies ( x cos y ), and

Let ( y tan^{-1}x ) which implies ( x tan y ).

Therefore, we can equate the expressions:

[ cos y tan y ]

Multiplying both sides by ( cos y ) gives:

[ cos^2 y sin y Rightarrow 1 - sin^2 y sin y Rightarrow sin^2 y sin y - 1 0 ]

This is a quadratic equation in terms of ( sin y ). Solving for ( sin y ), we get:

[ sin y frac{-1 pm sqrt{1 4}}{2} frac{-1 pm sqrt{5}}{2} ]

Since the sine function can only have values between -1 and 1, we discard the negative root. Thus:

[ sin y frac{-1 sqrt{5}}{2} ]

To find ( cos^2 y ), we use the identity ( 1 - sin^2 y cos^2 y ):

[ cos^2 y 1 - left( frac{-1 sqrt{5}}{2} right)^2 frac{-1 sqrt{5}}{2} ]

Since ( cos y ) can be positive or negative, we have:

[ cos y pm sqrt{frac{-1 sqrt{5}}{2}} ]

Thus, the value of ( x ) is:

[ x pm sqrt{frac{-1 sqrt{5}}{2}} approx 0.786151 ]

However, for the equation ( cos^{-1}x tan^{-1}x ) to hold true, ( x ) must be positive, since ( arccos x ) and ( arctan x ) are defined in the first and second quadrants, respectively. Therefore:

[ x sqrt{frac{-1 sqrt{5}}{2}} approx 0.786151 ]

Verification: Plugging this value into a calculator confirms the solution.

What is ( cos(tan^{-1}x) )?

This is a different but related problem. Let ( theta tan^{-1}x ). Then ( x tan theta ). We can represent this using a right triangle where the opposite side is ( x ) and the adjacent side is 1. The hypotenuse is then ( sqrt{1 x^2} ).

[ cos(tan^{-1}x) costheta frac{text{adj}}{text{hyp}} frac{1}{sqrt{1 x^2}} ]

This shows the direct relationship between the tangent and cosine functions in a right triangle.

Concluding Remarks

Understanding and solving inverse trigonometric equations, such as ( cos^{-1}x tan^{-1}x ), not only deepens our knowledge of trigonometry but also enhances problem-solving skills in mathematics. Whether you're a student or a professional, these concepts are fundamental in various fields, including physics and engineering.

Remember, the key to mastering inverse trigonometric functions lies in practice and understanding the underlying geometric and algebraic relationships.