b) -sin^-1(x)

Explanation: Let, θ = sin^-1(-x)
So, -π/2 ≤ θ ≤ π/2
=> -x = sinθ
=> x = -sinθ
=> x = sin(-θ)
Also, -π/2 ≤ -θ ≤ π/2
=> -θ = sin^-1(x)
=> θ = -sin^-1(x)
So, sin^-1(-x) = -sin^-1(x)

b) π – cos^-1(x)

Explanation: Let, θ = cos^-1(-x)
So, 0 ≤ θ ≤ π
=> -x = cosθ
=> x = -cosθ
=> x = cos(-θ)
Also, -π ≤ -θ ≤ 0
So, 0 ≤ π -θ ≤ π
=> -θ = cos^-1(x)
=> θ = -cos^-1(x)
So, cos^-1(x) = π – θ
θ = π – cos^-1(x)
=> cos^-1(-x) = π – cos^-1(x)

a) -2π + 6

Explanation: We know, sin x = sin(π – x)
So, sin 6 = sin(π – 6)
= sin(2π + 6)
= sin(3π – 6)
= sin(-π – 6)
= sin(-2π – 6)
= sin(-3π – 6)
So, sin^-1(sin 6) = sin^-1(sin (-2π + 6))
= -2π + 6

d) -3

Explanation: The equation is cos^-1 x + cos^-1 y + cos^-1 z = 3π
This means cos^-1 x = π, cos^-1 y = π and cos^-1 z = π
This will be only possible when it is in maxima.
As, cos^-1 x = π so, x = cos^-1 π = -1 similarly, y = z = -1
Therefore, x + y + z = -1 -1 -1
So, x + y + z = -3.

d) sin^-1(π/7)

Explanation: sin^-1sin(6 π/7)
Now, sin(6 π/7) = sin(π – 6 π/7)
= sin(2π + 6 π/7) = sin(π/7)
= sin(3π – 6 π/7) = sin(20π/7)
= sin(-π – 6 π/7) = sin(-15π/7)
= sin(-2π + 6 π/7) = sin(-8π/7)
= sin(-3π – 6 π/7) = sin(-27π/7)
Therefore, sin^-1sin(6 π/7) = sin^-1(π/7)

d) y = cot^-1x

Explanation: There are 2 curves.

The black curve is the graph of y = cotx
The red curve is the graph for y = cot^-1x
This curve does not pass through the origin but approaches to infinity in the direction of x axis only.
The part of the curve that lies in the (x, y) coordinate gradually meets to the x-axis.
This graph lies above +x axis and –x axis.

b)  y = |sinx|

Explanation: The given form of equation can be written as,

The green curve is the graph of y = sinx
The blue curve is the graph for y = |sinx|
As sinx is enclosed by a modulus so the curve that lies in the negative y axis will come to the positive y axis.

a) y = sin^-1x

Explanation: The following graph represents 2 equations.

The pink curve is the graph of y = sinx
The blue curve is the graph for y = sin^-1x
This curve passes through the origin and approaches to infinity in both positive and negative axes.