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TOPICS
Unit-2(Examples)

Example-1 :-  Find the principal value of sin⁻¹(1/√2).

Solution :-
```   Let sin⁻¹(1/√2) = θ.
sin θ = 1/√2
sin θ = sin(π/4)  {Range = [-π/2, π/2]}
θ = π/4
The principal value of sin⁻¹(1/√2) = π/4
```

Example-2 :-  Find the principal value of cot⁻¹(-1/√3).

Solution :-
```   Let cot⁻¹(-1/√3) = θ.
cot θ = -1/√3
cot θ = -cot(π/3)  {Range = [0, π]}
cot θ = cot(π - π/3)
cot θ = cot(2π/3)
θ = 2π/3
The principal value of cot⁻¹(-1/√3) = 2π/3
```

Example-3 :-  Show that
(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x
(ii) sin⁻¹(2x√1 - x²) = 2cos⁻¹x.

Solution :-
```(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x
Let x = sin θ, then θ = sin⁻¹x
L.H.S
sin⁻¹(2x√1 - x²)
= sin⁻¹(2 sin θ √1 - sin² θ)        ∴[√1 - sin² θ = cos θ]
= sin⁻¹(2 sin θ cos θ)    ∴[sin 2θ = 2 sin θ cos θ]
= sin⁻¹(sin 2θ)
= 2θ
= 2 sin⁻¹x
= R.H.S

(ii)sin⁻¹(2x√1 - x²) = 2cos⁻¹x.
Let x = cos θ, then θ = cos⁻¹x
L.H.S
sin⁻¹(2x √1 - x²)
= sin⁻¹(2 sin θ √1 - sin² θ)        ∴[√1 - sin² θ = cos θ]
= sin⁻¹(2 sin θ cos θ)    ∴[sin 2θ = 2 sin θ cos θ]
= sin⁻¹(sin 2θ)
= 2θ
= 2 cos⁻¹x
= R.H.S
```

Example-4 :-  Show that

Solution :-
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Example-5 :-  Express , -3π/2 < x < π/2 in the simplest form.

Solution :-
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Example-6 :-  Write , x > 1 in the simplest form.

Solution :-
```
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Example-7 :-  Prove that

Solution :-
```
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Example-8 :-  Find the value of cos(sec⁻¹x + cosec⁻¹x), | x | ≥ 1.

Solution :-
```   cos(sec⁻¹x + cosec⁻¹x)
∴[sec⁻¹x + cosec⁻¹x = π/2]
= cos(π/2)
= 0
```
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