# KC Sinha Solution Class 12 Chapter 4 Inverse Trigonometric Function Exercise 4.1

## Exercise 4.1

TYPE 1

#### Question 1

Find the value of

(i)

Sol :

Let ..(i)

..(ii)

On comparing (i) and (ii) equations

(ii)

Sol :

⇒Let ..(i)

..(ii)

From (i) and (ii) , we get

(iii)

Sol :

Note: cot-1 (-θ)=π-cot-1 θ

⇒cot-1 (-√3)=π-cot-1 (√3)

(vi)

Sol :

Let ..(i)

∵[cot(π+θ)=cotθ]

..(ii)

From (i) and (ii) , we get

(v)

Sol :

Let ..(i)

[∵ tan(π-θ)=-tanθ]

..(ii)

From (i) and (ii) , we get

(vi)

Sol :

Note :

(vii)

Sol :

Let ..(i)

[∵ tan(π+θ)=tanθ]

..(ii)

From (i) and (ii) , we get

(viii)

Sol :

Let ..(i)

[∵ cos(π+θ)=-cosθ]

..(ii)

From (i) and (ii) , we get

(ix)

Sol :

Let ..(i)

[∵ sin(π-θ)=sinθ]

..(ii)

From (i) and (ii)

#### Question 2

(i) tan-1 (√3)

Sol :

Let tan-1 (√3)=θ

⇒tanθ=√3

(ii)

Sol :

Let

(iii) tan-1 (-1)

Sol :

Let tan-1 (-1)=θ

⇒tanθ=-1

(iv) cosec-1 (2)

Sol :

Let cosec-1 (2)=θ

⇒cosecθ=2

(v)

Sol :

Let

(vi)

Sol :

Let

(vii)

Sol :

Let ,

,

,

[∵ sin(π-θ)=sinθ]

⇒α+β

(viii)

Sol :

[∵ sin-1 (-x)=-sin-1 x]

#### Question 3

(i)

Sol :

Let

[h=√p2+b2

=√32+42

=√25=5]

On putting in the place of

We get

(ii)

Sol :

(iii)

Sol :

TYPE 2

#### Question 4

(i) |x|>1

Sol :

Putting x=secθ , then θ=sec-1 x

Now ,

⇒tan-1 cotθ

(ii) , x≠0

Sol :

Putting x=tanθ and θ=tan-1 x

Now

(iii)

Sol :

Note:

(iv)

Sol :

Note :

Now

(v)

Sol :

Putting x=tanθ , then θ=tan-1 x

Now

[∵ 1+tan2θ=sec2θ]

(vi)

Sol :

Putting x=cosθ , then

Now ,

[dividing by cosθ]

TYPE 3

#### Question 5

Prove that:

(i)

Sol :

Note:

Now , L.H.S =

=R.H.S

Hence Proved

(ii)

Sol :

Note :

Now L.H.S=

R.H.S..(i)

Again , L.H.S=

..(ii)

From (i) and (ii) , we get

Hence Proved

(iii)

Sol :

L.H.S =

=R.H.S

Hence Proved

(iv)

Sol :

L.H.S=

=R.H.S proved

(v)

Sol :

Now, L.H.S

R.H.S proved

(vi)

Sol :

Note :

Now , L.H.S =

R.H.S proved

(vii) tan-1 1+tan-1 2+tan-1 3=π

Sol :

L.H.S =tan-1 1+tan-1 2+tan-1 3

=

=

=

=tan-1 3 – tan-1 3

=

=

=tan-1(tanπ)=π=R.H.S..(i)

Again , R.H.S=

=

=

=

=

=

=

=

= L.H.S ..(ii)

From (i) and (ii)

⇒tan-1 1+tan-1 2+tan-1 3=π

(viii)

Sol :

Given ,

=

=

=

= R.H.S

proved

(ix)

Sol :

Let and

Now , L.H.S=

= R.H.S

proved

(x)

Sol :

Let

⇒8 tanθ=3(1+tan2 θ)

⇒8 tanθ=3+3tan2 θ)

⇒3tan2 θ-8tanθ+3=0

Let tanθ=x , then 3x2-8x+3=0

Discriminant (D)=b2-4ac

=(-8)2-4×3×3

=64-36=28>0

or

R.H.S

Proved

#### Question 6

Prove that

(i)

Sol :

L.H.S=tan-1 x + cot-1 y

=

=

=

= R.H.S

(ii) tan-1 x + cot-1(1+x) = tan-1(1+x+x2)

Sol :

L.H.S= tan-1 x+cot-1 (1+x)

=

=

=

=

= tan-1(1+x+x2) R.H.S

Proved

(iii)

Sol :

L.H.S=

=

=

=

=

=

=R.H.S

proved

(iv) 2 cot-1 5+cot-1 7+2 cot-18=π/4

Sol :

L.H.S=2 cot-1 5 + cot-17+2 cot-18

=2 cot-1 5 +2 cot-18+ cot-17

=2(cot-1 5+cot-1 8)+cot-1 7

=

=

=

=

=

=

=

=

=

= R.H.S

Proved

(v)

Sol :

L.H.S=

=

=

=

=

= R.H.S

#### Question 7

Prove that :

(i) ab>-1 , bc>-1 , ca>-1

Sol :

L.H.S=

=tan-1 a-tan-1 b+tan-1 b-tan-1 c+tan-1 c-tan-1 a

=tan-1 a-tan-1 a-tan-1 b+tan-1 b-tan-1 c+tan-1 c

=0 =R.H.S

proved

(ii)

Sol :

L.H.S=

=tan-1 a3-tan-1 b3+tan-1 b3-tan-1 c3+tan-1 c3-tan-1 a3

=tan-1 a3-tan-1 a3-tan-1 b3+tan-1 b3-tan-1 c3+tan-1 c3

=0 R.H.S

proved

#### Question 8

(i)

Sol :

L.H.S

*** QuickLaTeX cannot compile formula:
=sin^{-1} \left[\dfracc{3}{5}.\sqrt{1-\dfrac{8^2}{17^2}}+\dfrac{8}{17}.\sqrt{1-\left(\dfrac{3}{5}\right)^2}\right]

*** Error message:
Undefined control sequence \dfracc.
leading text: $=sin^{-1} \left[\dfracc  *** QuickLaTeX cannot compile formula: =sin^{-1} \left[\dfracc{3}{5} \times \sqrt{\dfrac{17^2-8^2}{17^2}}+\dfrac{8}{17} \times \sqrt{\dfrac{5^2-3^2}{5^2}}\right] *** Error message: Undefined control sequence \dfracc. leading text:$=sin^{-1} \left[\dfracc


*** QuickLaTeX cannot compile formula:
=sin^{-1} \left[\dfracc{3}{5} \times \dfrac{\sqrt{289-64}}{17}+\dfrac{8}{17} \times \dfrac{\sqrt{25-9}}{5}\right]

*** Error message:
Undefined control sequence \dfracc.
leading text: $=sin^{-1} \left[\dfracc  *** QuickLaTeX cannot compile formula: =sin^{-1} \left[\dfracc{3}{5} \times \dfrac{\sqrt{225}}{17}+\dfrac{8}{17} \times \dfrac{\sqrt{16}}{5}\right] *** Error message: Undefined control sequence \dfracc. leading text:$=sin^{-1} \left[\dfracc


*** QuickLaTeX cannot compile formula:
=sin^{-1} \left[\dfracc{3}{5} \times \dfrac{15}{17}+\dfrac{8}{17} \times \dfrac{4}{5}\right]

*** Error message:
Undefined control sequence \dfracc.
leading text: $=sin^{-1} \left[\dfracc  =R.H.S proved (ii) Sol : L.H.S *** QuickLaTeX cannot compile formula: =cos^{-1} \left(\dfrac{1008}{65\times 65}-\dfrac{\sqrt{81\times 49}}{65} \times \dfrac{\sqrt{128\times 2}{65}} \right) *** Error message: Missing } inserted. leading text: ...imes \dfrac{\sqrt{128\times 2}{65}} \right  R.H.S Proved (iii) Sol : L.H.S Let sin^{-1} \dfrac{77}{85}=\thetasin \theta=\dfrac{77}{85}tan \theta =\dfrac{77}{36}\theta = tan^{-1} \dfrac{77}{36}sin^{-1} \dfrac{77}{85}= tan^{-1} \dfrac{77}{36} *** QuickLaTeX cannot compile formula: + R.H.S proved <hr /> <strong>(iv) </strong> *** Error message: Missing$ inserted.



cos^{-1} \dfrac{12}{13}+sin^{-1} \dfrac{5}{13}=sin^{-1} \dfrac{56}{65}

*** QuickLaTeX cannot compile formula:
question may be incorrect

<hr />

<strong>(v) </strong>

*** Error message:
Missing $inserted.  cos^{-1} \dfrac{4}{5}+cos^{-1} \dfrac{12}{13}=cos^{-1} \dfrac{33}{65} *** QuickLaTeX cannot compile formula: Sol : L.H.S *** Error message: Missing$ inserted.



=cos^{-1} \dfrac{4}{5}+cos^{-1} \dfrac{12}{13}=cos^{-1} \left(\dfrac{4}{5} \times \dfrac{12}{13} – \sqrt{1-\dfrac{16}{25} \times \sqrt{1-\dfrac{144}{169}}\right)=cos^{-1} \left( \dfrac{48}{65}-\dfrac{3}{5} \times \dfrac{5}{13}\right)=cos^{-1} \left(\dfrac{48}{65}-\dfrac{15}{65}\right)=cos^{-1} \dfrac{33}{65}

*** QuickLaTeX cannot compile formula:
= R.H.S proved

<hr />

<strong>(vi) </strong>

*** Error message:
Missing $inserted.  sin^{-1} \dfrac{3}{5}-sin^{-1} \dfrac{8}{17}=cos^{-1} \dfrac{84}{85} *** QuickLaTeX cannot compile formula: Sol : L.H.S *** Error message: Missing$ inserted.



sin^{-1} \dfrac{3}{5}-sin^{-1} \dfrac{8}{17}=sin^{-1} \left(\dfrac{3}{5} \times \sqrt{1-\left(\dfrac{8}{17}\right)^2}-\dfrac{8}{17} \times \sqrt{1-\left(\dfrac{9}{5}\right)^2}\right)\$

working

(vii)

(viii)

(ix)

(x)

(xi)

#### Question 19

later

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