Prove the following
Question 1. 3sin-1x = sin-1(3x – 4x3), x∈[-1/2, 1/2]
Solution:
Let us take x = sinθ, so θ = sin-1x
Substitute the value of x in the equation present on R.H.S.
The equation becomes sin-1(3sinθ – 3sin3θ)
We know, sin3θ = 3sinθ – 4sin3θ
So , sin-1(3sinθ – 3sin3θ) = sin-1(sin3θ)
By the property of inverse trigonometry we know, sin(sin-1(θ)) = θ
So, sin-1(sin3θ) = 3θ
And we know θ = sin-1x
So, 3θ = 3sin-1x = L.H.S
Question 2. 3cos-1x = cos-1(4x3 – 3x), x∈[-1/2, 1]
Solution:
Let us take x = cosθ, so θ = cos-1x
Substitute value of x in the equation present on R.H.S.
The equation becomes cos-1(4cos3θ – 3cosθ)
We know, cos3θ = 4cos3θ – 3cosθ
So, cos-1(4cos3θ – 3cosθ) = cos-1(cos3θ)
By the property of inverse trigonometry we know, cos(cos-1(θ)) = θ
So, cos-1(cos3θ) = 3θ
And we know θ = cos-1x
So, 3θ = 3cos-1x = L.H.S
Write the following functions in simplest forms:
Question 3. [Tex]tan^{-1}\frac{\sqrt{1+x^2}-1}{x} , x \neq 0 [/Tex]
Solution:
Let us assume that x = tanθ, so θ = tan-1x
Substitute the value of x in question.
So equation becomes [Tex]tan^{-1}\frac{\sqrt{1+tan^2θ}-1}{tanθ}[/Tex]
We know that, 1 + tan2θ = sec2θ
Replacing 1 + tan2θ with sec2θ in the equation [Tex]tan^{-1}\frac{\sqrt{1+tan^2θ}-1}{tanθ}[/Tex]
So equation becomes, [Tex]tan^{-1}\frac{\sqrt{sec^2θ}-1}{tanθ}[/Tex]
[Tex]= tan^{-1}\frac{secθ-1}{tanθ}[/Tex]
We know, tanθ = sinθ/cosθ and sec = 1/cosθ
Replacing value of tanθ and secθ in [Tex] tan^{-1}\frac{secθ-1}{tanθ}[/Tex]
[Tex]= tan^{-1}\frac{\frac{1}{cosθ}-1}{\frac{sinθ}{cosθ}}[/Tex]
[Tex]= tan^{-1}\frac{\frac{1-cosθ}{cosθ}}{\frac{sinθ}{cosθ}}[/Tex]
[Tex]= tan^{-1}\frac{1-cosθ}{sinθ}[/Tex]
We know, 1 – cosθ = 2sin2θ/2​ and sinθ = 2sinθ/2cosθ/2
So the equations after replacing above value becomes [Tex]tan^{-1}\frac{2sin^2\frac{θ}{2}}{2sin\frac{θ}{2}cos\frac{θ}{2}}[/Tex]
[Tex]= tan^{-1}\frac{sin\frac{θ}{2}}{cos\frac{θ}{2}}[/Tex]
We know [Tex]\frac{\frac{sinθ}{2}}{\frac{cosθ}{2}} = tan\frac{θ}{2} [/Tex]
[Tex]= tan^{-1}tan\frac{θ}{2}[/Tex]
= θ/2 [tan-1(tanθ) = θ]
= 1/2 tan-1x [θ = tan-1x]
Question 4. [Tex]tan^{-1}(\sqrt{\frac{1-cosx}{1+cosx}}), 0<x<\pi[/Tex]
Solution:
We know, 1 – cosx = 2sin2x/2 and 1 + cosx = 2cos2x/2
Substituting above formula in question
[Tex]=tan^{-1}(\sqrt{\frac{2sin^2\frac{x}{2}}{2cos^2\frac{x}{2}}})[/Tex]
[Tex]=tan^{-1}(\frac{sin\frac{x}{2}}{cos\frac{x}{2}})[/Tex]
= tan-1(tanx/2) [Tex][\frac{sin\theta}{cos\theta} = tan\theta][/Tex]
= x/2 [tan-1(tanθ) = θ]
Question 5. [Tex]tan^{-1}(\frac{cosx-sinx}{cosx+sinx}), \frac{-\pi}{4}<x<\frac{3\pi}{4}[/Tex]
Solution:
Divide numerator and denominator by [Tex]cosx[/Tex]
[Tex]= tan^{-1}(\frac{\frac{cosx-sinx}{cosx}}{\frac{cosx+sinx}{cosx}})[/Tex]
[Tex]= tan^{-1}(\frac{\frac{cosx}{cosx}-\frac{sinx}{cosx}}{\frac{cosx}{cosx}+\frac{sinx}{cosx}})[/Tex]
[Tex]= tan^{-1}(\frac{1-\frac{sinx}{cosx}}{1+\frac{sinx}{cosx}})[/Tex]
We know, [Tex]\frac{sinx}{cosx} = tanx[/Tex]
[Tex]= tan^{-1}(\frac{1-tanx}{1+tanx})[/Tex]
This can also be written as [Tex]tan^{-1}(\frac{1-tanx}{1+1.tanx}) [/Tex] – (1)
We know [Tex]tan^{-1}x-tan^{-1}y = tan^{-1}\frac{x-y}{1+x.y} [/Tex] – (2)
On comparing equation (1) and (2) we can say that x = 1 and y = tan-1x
So we can say that [Tex]tan^{-1}(\frac{1-tanx}{1+1.tanx}) = tan^{-1}1 – tan^{-1}x[/Tex]
= Ï€/4​ – tan−1x [tan−11 = Ï€/4​]
Question 6. [Tex]tan^{-1}\frac{x}{\sqrt{a^2-x^2}} , |x|<a[/Tex]
Solution:
Let us assume that x = asinθ, so θ = sin -1x/a
Substitute the value of x in question.
[Tex]= tan^{-1}\frac{asin\theta}{\sqrt{a^2-(asin\theta)^2}}[/Tex]
[Tex]= tan^{-1}\frac{asin\theta}{\sqrt{a^2-a^2sin^2\theta}}[/Tex]
Taking a2 common from denominator
[Tex]= tan^{-1}\frac{asin\theta}{\sqrt{a^2(1-sin^2\theta)}}[/Tex]
We know that, sin2θ + cos2θ = 1, so 1 – sin2θ = cos2θ
[Tex]= tan^{-1}\frac{asin\theta}{\sqrt{a^2cos^2\theta}}[/Tex]
[Tex]= tan^{-1}\frac{asin\theta}{acos\theta}[/Tex]
[Tex]= tan^{-1}\frac{sin\theta}{cos\theta}[/Tex]
= tan-1(tanθ) [sinθ/cosθ = tanθ]
= θ
= sin-1x/a
Question 7. [Tex]tan^{-1}(\frac{3a^2x-x^3}{a^3-3ax^2}),a>0;\frac{-a}{\sqrt{3}}<x<\frac{a}{\sqrt{3}}[/Tex]
Solution:
Let us assume that x = atanθ, so θ = tan -1x/a
Substitute the value of x in question
[Tex]= tan^{-1}(\frac{3a^2(atan\theta)-(atan\theta)^3}{a^3-3a(atan\theta)^2})[/Tex]
[Tex]= tan^{-1}(\frac{3a^3tan\theta-a^3tan^3\theta}{a^3-3a^3tan^2\theta})[/Tex]
Taking a3common from numerator and denominator
[Tex]= tan^{-1}(\frac{a^3(3tan\theta-tan^3\theta)}{a^3(1-3tan^2\theta)})[/Tex]
[Tex]= tan^{-1}(\frac{3tan\theta-tan^3\theta}{1-3tan^2\theta})[/Tex]
We know [Tex]tan3\theta = \frac{3tan\theta-tan^3\theta}{1-3tan^2\theta}[/Tex]
So, [Tex]tan^{-1}(\frac{3tan\theta-tan^3\theta}{1-3tan^2\theta})= tan^{-1}(tan3\theta)[/Tex]
= 3θ [ tan-1(tanθ) = θ]
= 3tan -1x/a
Find the values of each of the following:
Question 8. tan−1[2cos(2sin−11/2​)]
Solution:
Let us assume that sin−11/2 = x
So, sinx = 1/2
Therefore, x = π​/6 = sin−11/2
Therefore, tan−1[2cos(2sin−11/2​)] = tan−1[2cos(2 * π​/6)]
= tan−1[2cos(π​/3)]
Also, cos(π/3​) = 1/2​
Therefore, tan−1[2cos(π​/3)] = tan−1[(2 * 1/2)]
= tan−1[1] = π​/4
Question 9. [Tex]tan\frac{1}{2}[sin^{-1}\frac{2x}{1+x^2}+cos^{-1}\frac{1-y^2}{1+y^2}],|x|<1,y>0,xy<1[/Tex]
Solution:
We know, 2tan-1x = [Tex]sin^{-1}\frac{2 x}{1+x^2} [/Tex] and 2tan-1y = [Tex]cos^{-1}[\frac{1 – y^2 }{1+y^2}][/Tex]
[Tex]\therefore tan\frac{1}{2}[sin^{-1}\frac{2x}{1+x^2}+cos^{-1}\frac{1-y^2}{1+y^2}] [/Tex]
= tan(1/2)​[2(tan−1x + tan−1y)]
= tan[tan−1x + tan−1y]
Also, tan−1x + tan−1y = [Tex]tan^{-1}\frac{x+y}{1-xy}[/Tex]
Therefore, tan[tan−1x + tan−1y] = [Tex]tan[tan^{-1}\frac{x+y}{1-xy}][/Tex]
= (x + y)/(1 – xy)
Question 10. sin − 1(sin2π/3​)
Solution:
We know that sin−1(sinθ) = θ when θ ∈ [-π/2, π/2], but [Tex]\frac{2 \pi}{3} > \frac{\pi}{2}[/Tex]
So, sin − 1(sin2π/3​) can be written as [Tex]sin^{-1}[sin(\pi-\frac{2\pi}{3})][/Tex]
sin − 1(sinπ/3​) here [Tex]\frac{-\pi}{2}<\frac{\pi}{3}<\frac{\pi}{2}[/Tex]
Therefore, sin − 1(sinπ/3​) = π/3
Question 11. tan−1(tan3π/4​)
Solution:
We know that tan−1(tanθ) = θ when [Tex]\theta \epsilon(\frac{-\pi}{2},\frac{\pi}{2}) [/Tex] but [Tex]\frac{3 \pi}{4} > \frac{\pi}{2}[/Tex]
So, tan−1(tan3π/4​) can be written as tan−1(-tan(-3π/4)​)
= tan−1[-tan(Ï€ – Ï€/4​)]
= tan−1[-tan(π/4​)]
= –tan−1[tan(Ï€/4​)]
= – Ï€/4 where [Tex]\frac{-\pi}{4} \epsilon(\frac{-\pi}{2},\frac{\pi}{2})[/Tex]
Question 12. [Tex]tan(sin^{-1}\frac{3}{5} + cot^{-1}\frac{3}{2})[/Tex]
Solution:
Let us assume [Tex]sin^{-1}\frac{3}{5} [/Tex] = x , so sinx = 3/5
We know, [Tex]cosx = \sqrt{1-sin^2x}[/Tex]
[Tex]\therefore cosx = \sqrt{1-(\frac{3}{5})^2}[/Tex]
[Tex]cosx = \sqrt{1-\frac{9}{25}}[/Tex]
[Tex]cosx = \sqrt{\frac{25-9}{25}}[/Tex]
[Tex]cosx = \sqrt{\frac{16}{25}}[/Tex]
cosx = 4/5
We know, [Tex]tanx = \frac{sinx}{cosx}[/Tex]
So, [Tex]tanx = \frac{\frac{3}{5}}{\frac{4}{5}}[/Tex]
tanx = 3/4
Also, [Tex]tan^{-1}\frac{1}{x} = cot^{-1}x[/Tex]
Hence, [Tex]tan(sin^{-1}\frac{3}{5} + cot^{-1}\frac{3}{2}) = tan(tan^{-1}\frac{3}{4}+tan^{-1}\frac{2}{3})[/Tex]
tan-1x + tan-1y = [Tex]tan^{-1}\frac{x+y}{1-xy}[/Tex]
So, [Tex]tan(tan^{-1}\frac{3}{4}+tan^{-1}\frac{2}{3}) = tan(tan^{-1}\frac {\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}.\frac{2}{3}})[/Tex]
[Tex]= tan(tan^{-1}\frac{\frac{9+8}{12}}{\frac{12-6}{12}})[/Tex]
[Tex]= tan(tan^{-1}\frac{17}{6})[/Tex]
= 17/6
Question 13. cos−1(cos7π/6​) is equal to
(i) 7Ï€/6 (ii) 5Ï€/6 (iii)Ï€/3 (iv)Ï€/6
Solution:
We know that cos−1(cosθ) = θ, θ ∈ [0, π]
cos−1(cosθ) = θ, θ ∈ [0, π]
Here, 7π/6 > π
So, cos−1(cos7π/6​) can be written as cos−1(cos(-7π/6)​)
= cos−1[cos(2Ï€ – 7Ï€/6​)] [cos(2Ï€ + θ) = θ]
= cos−1[cos(5π/6​)] where 5π/6 ∈ [0, π]
Therefore, cos−1[cos(5π/6​)] = 5π/6
Question 14. [Tex]sin[\frac{\pi}{3} – sin^{-1}(-\frac{1}{2} )][/Tex]
(i) 1/2 (ii) 1/3 (iii) 1/4 (iv) 1
Solution:
Let us assume sin-1(-1/2)= x, so sinx = -1/2
Therefore, x = -π/6​
Therefore, sin[Ï€/3​ – (-Ï€/6​)]
= sin[π/3​ + (π/6​)]
= sin[3Ï€/6]
= sin[Ï€/2]
= 1
Question 15. [Tex]tan^{-1}\sqrt{3} – cot^{-1}(-\sqrt{3}) [/Tex] is equal to
(i) π (ii) -π/2 (iii)0 (iv)2√3
Solution:
We know, cot(−x) = −cotx
Therefore, tan-13 – cot-1(-3) = tan-13 – [-cot-1(3)]
= tan-13 + cot-13
Since, tan-1x + cot-1x = π/2
Tan-13 + cot-13 = -Ï€/2
Deleted Questions
[Tex] tan^{-1}\frac{2}{11} + tan^{-1}\frac{7}{24} = tan^{-1}\frac{1}{2} [/Tex]
Solution:
We know, [Tex] tan^{-1}x + tan^{-1}y = tan^{-1}\frac{x+y}{1-xy} [/Tex]
Now put x = 2/11 and y = 7/24
So, [Tex] tan^{-1}\frac{2}{11} + tan^{-1}\frac{7}{24} = tan^{-1}\frac{\frac{2}{11}+\frac{7}{24}}{1-\frac{2}{11}.\frac{7}{24}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{48}{264}+\frac{77}{264}}{1-\frac{14}{264}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{48+77}{264}}{\frac{264-14}{264}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{48+77}{264}}{\frac{264-14}{264}} [/Tex]
[Tex]=tan^{-1}\frac{125}{250} [/Tex]
[Tex]=tan^{-1}\frac{1}{2} [/Tex]
= R.H.S
[Tex] 2tan^{-1}\frac{1}{2} + tan^{-1}\frac{1}{7} = tan^{-1}\frac{31}{17} [/Tex]
Solution:
We have to first write 2tan-1x in terms of tan-1x
We know that 2tan-1x = [Tex]tan^{-1}\frac{2x}{1-x^2} [/Tex]
Put x = 1/2 in the above formula
So, [Tex]2tan^{-1}\frac{1}{2} = tan^{-1}\frac{2.\frac{1}{2}}{1-(\frac{1}{2})^2}[/Tex]
[Tex]= tan^{-1}\frac{1}{1-(\frac{1}{4})^2}[/Tex]
[Tex]= tan^{-1}\frac{1}{\frac{4-1}{4}}[/Tex]
[Tex]= tan^{-1}\frac{1}{\frac{3}{4}}[/Tex]
[Tex]= tan^{-1}\frac{4}{3}[/Tex]
Now we can replace [Tex]2tan^{-1}\frac{1}{2} [/Tex]with [Tex] tan^{-1}\frac{4}{3}[/Tex]
So equation in L.H.S become [Tex] tan^{-1}\frac{4}{3} + tan^{-1}\frac{1}{7}[/Tex]
We know , [Tex] tan^{-1}x + tan^{-1}y = tan^{-1}\frac{x+y}{1-xy} [/Tex]
So, [Tex] tan^{-1}\frac{4}{3} + tan^{-1}\frac{1}{7} = tan^{-1}\frac{\frac{4}{3}+\frac{1}{7}}{1-\frac{4}{3}.\frac{1}{7}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{28}{21}+\frac{3}{21}}{1-\frac{4}{21}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{28+3}{21}}{\frac{21-4}{21}} [/Tex]
[Tex]=tan^{-1}\frac{\frac{31}{21}}{\frac{17}{21}} [/Tex]
[Tex]=tan^{-1}\frac{31}{17} [/Tex]
= R.H.S
[Tex]tan^{-1}\frac{1}{\sqrt{x^2-1}} [/Tex] , |x| > 1
Solution:
Let us assume that x = cosecθ, so θ = cosec -1x
Substitute the value of x in question with [Tex]cosecθ [/Tex]
[Tex]=tan^{-1}\frac{1}{\sqrt{cosec^2θ-1}}[/Tex]
We know that, 1 + cot2θ = cosec2θ, so cosec2θ = 1 – cot2θ
[Tex]=tan^{-1}\frac{1}{\sqrt{cot^2θ}}[/Tex]
[Tex]=tan^{-1}\frac{1}{cotθ}[/Tex]
= tan-1(tanθ) [1/cotθ = tanθ]
= θ [tan-1(tanθ) = θ]
= cosec−1x [θ = cosec−1x]
= π/2 ​- sec−1x [cosec−1x + sec−1x = π/2​]
cot(tan−1a + cot−1a)
Solution:
We know, tan−1x + cot−1x = π​/2
Therefore, cot(tan−1a + cot−1a) = cot(π​/2) =0
If sin(sin−11/5​ + cos−1x) = 1 then find the value of x
Solution:
sin−11/5​ + cos−1x = sin−11
We know, sin−11 = π/2
Therefore, sin−11/5​ + cos−1x = π/2
sin−11/5​ = Ï€/2 – cos−1x
Since, sin−1x​ + cos−1x = π/2
Therefore, Ï€/2 – cos−1x = sin−1x
sin−11/5​ = sin−1x
So, x = 1/5
If [Tex]tan^{-1}\frac{x-1}{x-2} + tan^{-1}\frac{x+1}{x+2} = \frac{\pi}{4} [/Tex] , then find the value of x
Solution:
We know, tan−1x + tan−1y = [Tex]tan^{-1}\frac{x+y}{1-xy}[/Tex]
[Tex]tan^{-1}\frac{x-1}{x-2} + tan^{-1}\frac{x+1}{x+2} =tan^{-1}\frac{\frac{x-1}{x-2}+\frac{x+1}{x+2}}{1-\frac{x-1}{x-2}.\frac{x+1}{x+2}} = \frac{\pi}{4}[/Tex]
[Tex]tan^{-1}\frac{\frac{(x-1)(x+2)+(x+1)(x-2)}{(x-2)(x+2)}}{\frac{(x-2)(x+2)-(x+1)(x-1)}{(x-2)(x+2)}} = \frac{\pi}{4}[/Tex]
[Tex]tan^{-1}\frac{(x-1)(x+2)+(x+1)(x-2)}{(x-2)(x+2)-(x+1)(x-1)} = \frac{\pi}{4}[/Tex]
[Tex]tan^{-1}(\frac{x^2+2x-x-2+x^2-2x+x-2}{x^2-4-x^2+1}) = \frac{\pi}{4}[/Tex]
[Tex]tan^{-1}(\frac{x^2+x-2+x^2-x-2}{-3}) = \frac{\pi}{4}[/Tex]
[Tex]tan^{-1}(\frac{2x^2-4}{-3}) = \frac{\pi}{4}[/Tex]
[Tex]\frac{2x^2-4}{-3} = tan(\frac{\pi}{4})[/Tex]
[Tex]\frac{2x^2-4}{-3} = 1[/Tex]
2x2 – 4 = -3
2x2 – 4 + 3 = 0
2x2 – 1 = 0
x2 = 1/2
x = 1/√2, -1/√2
Find the values of each of the expressions in Exercises 16 to 18.
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