Inverse Trigonometric Functions

Relating f-1 (x) with f-1 (-x) and f-1 (1/x) will become easy if you understand the properties of trigonometry inverse functions. 

These properties include even and odd properties, multiplicative inverse (reciprocal), etc. In even and odd, the two trig functions become positive or negative with respect to values, while the reciprocal of inverse trigonometric functions is the multiplicative inverse. 

Therefore,

• The odd and even rule of trigonometry functions depends on the reflection and origin of the y-axis. For example, f-1(-x) = – f-1(x)

• The multiplicative inverse of the function is reciprocal. For instance, f-1(x) = f-1(1/x)

Before briefing the relation easily, knowing odd and even trigonometric functions are important.

Odd and Even Trigonometric Inverse Functions

Even and odd functions depend on the changes in terms of reflection or origin, i.e., 180 degrees. Even function is symmetric by reflection about the y-axis like:

f-1(-x) = f-1(x)  

[It becomes the same positive as a result but gets subtracted from π]

Whereas odd function is symmetric by 180 degrees rotation about the origin such as:

f-1(-x) = – f-1(x)

Relating f-1(x) with f-1(-x): Property Explanation with Proof

While applying odd and even rules to all inverse trigonometric functions, we find that arcsine, arccosecant, and arctangent are odd while arccosine, arcsecant, and arccotangent are even functions. 

In trigonometric functions, sin (-x) = – sin x and cos(-x) = cos x; but talking about inverse trigonometric functions,

• sin-1(-x) = -sin-1(x)

• cosec-1(-x) = -cosec-1(x)

• tan-1(-x) = – tan-1(x)

• cos-1(-x) = π – cos-1(x)

• sec-1(-x) = π – sec-1(x)

• cot-1(-x) = π – cot-1(x)

Proving cos-1(-x) = π – cos-1(x)

Suppose cos-1(-x) = y 

It means,

-x = cos y 

x = – cos y 

Cosine is even function, and it can be written f(-θ) = f(π – θ). So,

x = – cos y = cos (π – y)

Hence, applying inverse both sides on X = cos (π – y)

cos-1(x) = cos-1(cos (π – y))   [cos-1(cos x) = x]

cos-1(x) = π – y

You know that y = cos-1(-x), so replacing y gives,

cos-1(x) = π – cos-1(-x)

Thus, finding result for cos-1(-x),

cos-1(-x) = π – cos-1(x)

Similarly, when you use the same concept on other even trigonometric inverse functions, you obtain the following results: 

1. sec-1(-x) = π – sec-1(x) where |x| > 1

2. cot-1(-x) = π – cot-1(x) where x ∈ R

Proving sin-1(-x) = -sin-1(x)

Suppose y = sin-1(-x) 

Applying inverse sine both side gives,

sin y = – x

Multiplying ‘-1’ both sides,

x = – sin y

Sine is odd function and so it can be written as f(-θ) = -f(θ);

x = sin (-y)

Again, multiplying inverse sin both the sides,

sin-1x = sin-1(sin(-y))  [sin-1(sinx) = x]

sin-1x = -y

Replacing y = sin-1(-x) value here,

sin-1x = -y = – sin-1(-x)

Hence proved that

sin-1(-x) = – sin-1x

Similarly, using this same concept in remaining odd functions, the result will be: 

1. tan-1(-x) = – tan-1x

2. cosec-1(-x) = – cosec-1x

Therefore, in simple words, the relation between f-1(x) and f-1(-x) depends on the odd and even functions. 

1. If even, f-1(-x) = – f(x) 

2. If odd f-1(-x) = π – f(x)

Relating f-1(X) with f-1(1/X): Reciprocal Property Explained

The reciprocal of trigonometric inverse functions gives the opposite of inverse trigonometric functions. Reciprocal property of inverse trigonometry functions results as:

• sin-1(x) = cosec-1(1/x)

• cosec-1(x) = sin-1(1/x)

• cos-1(x) = sec-1(1/x)

• sec-1(x) = cos-1(1/x)

• tan-1(1/x) = – π + cot-1(x)

Proving sin-1(x) = cosec-1(1/x)

Let sin-1x = y

Multiplying sine both sides,

sin(sin-1x) = sin y    [Note: sin(sin-1x) = x]

x = sin y       …..(i)

Reciprocal on both sides,

1/x = 1/siny

We know that 1/sin y  = cosec y

1/x = cosec y

Multiplying inverse cosec both sides,

cosec-1(1/x) = cosec-1(cosec y)

cosec-1(1/x) = y

Replacing y = sin-1x value above

cosec-1(1/x) = y = sin-1x

Hence,

sin-1x = cosec-1(1/x)

Similarly, using the same concept, other inverse trigonometry results are obtained. 

Proving tan-1(1/x) = – π + cot-1x

Let cot-1x = y

Multiplying cot both sides,

x = cot y     ….(i)

Tan ( – π + y) = – tan( π- y)

Tan ( – π + y) = tan y  

Tan ( – π + y) = 1 / cot y

Tan ( – π + y) = 1/x

Multiplying inverse tan both sides,

 – π + y = tan-1(1/x)

Replacing y with cot-1x, we get 

tan-1(1/x) = – π + cot-1x

Therefore, in simple words, the relation between f-1(x) and f-1(1/x) is reciprocal with the opposite trigonometric functions. 

f-1(x) = f-1(1/x)

Conclusion

Relating f-1 (x) with f-1 (-x) and f-1 (1/x) needs the information of inverse trigonometry properties. Learning all properties of inverse trigonometric functions helps solve all the questions easily. With some tricks, tips, and properties, the solving steps become less, giving accurate results. 

Candidates planning to crack up entrance exams must study and keep all properties of inverse trigonometric functions in mind to evaluate problems quickly. Understanding the relation between f-1(x) with both f-1(-x) and f-1(1/x) is the even-odd function property and reciprocal property, respectively.