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sum(r =1)^(n) sin^(-1) ((sqrtr - sqrt(r...

`sum_(r =1)^(n) sin^(-1) ((sqrtr - sqrt(r -1))/(sqrtr(r + 1)))` is equal to

A

`tan^(-1) (sqrtn) - (pi)/(4)`

B

`tan^(-1) (sqrt(n+1)) - (pi)/(4)`

C

`tan^(-1)(sqrtn)`

D

`tan^(-1) (sqrt(n+1))`

Text Solution

Verified by Experts

The correct Answer is:
C
`sin^(-1) ((sqrtr - sqrt(r -1))/(sqrt(r( r+1)))) = tan^(-1) ((sqrtr - sqrt(r -1))/(1 + sqrt(r (r -1))))`
`:. underset(r =1)overset(n) sum sin^(-1) ((sqrtr - sqrt(r -1))/(sqrt(r (r + 1))))`
`= underset(r =1)overset(n) sum (tan^(-1) sqrtr - tan^(-1) sqrt(r -1)) = tan^(-1) sqrtn`
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CENGAGE-INVERSE TRIGONOMETRIC FUNCTIONS-Exercise (Single)
  1. Arithmetic mean of the non-zero solutions of the equation tan^-1 (1/(2...

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  2. If cot^(-1)x+cot^(-1)y+cot^(-1)z=pi/2,x , y , z >0a n dx y<1, then x+y...

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  3. If x^2+y^2+z^2=r^2,t h e ntan^(-1)((x y)/(z r))+tan^(-1)((y z)/(x r))+...

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  4. The value of tan^(-1)((xcostheta)/(1-xsintheta))-cot^(-1)((costheta)/(...

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  5. if cot^(-1)[sqrt(cosalpha)]-tan^(-1)[sqrt(cosalpha)]=x then sinx is eq...

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  6. sum(r =1)^(n) sin^(-1) ((sqrtr - sqrt(r -1))/(sqrtr(r + 1))) is equal...

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  7. sum(m=1)^(n) tan^(-1) ((2m)/(m^(4) + m^(2) + 2)) is equal to

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  8. The value of tan^(-1).(4)/(7) + tan^(-1).(4)/(19) + tan^(-1).(4)/(39) ...

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  9. The sum of series sec^(-1)sqrt(2)+sec^(-1)(sqrt(10))/3+sec^(-1)(sqrt(5...

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  10. If (1)/(2) sin^(-1) [(3 sin 2 theta)/(5 + 4 cos 2 theta)] = tan^(-1) x...

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  11. The value 2tan^(-1)[sqrt((a-b)/(a+b)tantheta/2)] is equal to cos^(-1)(...

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  12. If sin^(-1) ((2a)/(1+a^2))+ sin^(-1) ((2b)/(1+b^2)) = 2 tan^(-1)x then...

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  13. If 3\ sin^(-1)((2x)/(1+x^2))-4\ cos^(-1)((1-x^2)/(1+x^2))+2\ tan^(-1)(...

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  14. If x(1) = 2 tan^(-1) ((1 + x)/(1 -x)), x(2) = sin^(-1) ((1 - x^(2))/(1...

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  15. If the equation x^3+b x^2+c x+1=0,(b<c), has only one real rootalpha ,...

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  16. The value of sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x^2)] is equal to sin^...

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  17. If cos^(- 1)x-cos^(- 1)(y/2)=alpha then 4x^2-4xycosalpha+y^2=

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  18. If sin^(-1)x+sin^(-1)y+sin^(-1)z=pi,t h e nx^4+y^2+z^4+4x^2y^2z^2=K(x^...

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  19. If f(x)=sin^(-1)((sqrt(3))/2x-1/2sqrt(1-x^2)),-1/2lt=xlt=1,t h e nf(x)...

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  20. If 2^2pi//sin^((-1)x)-2(a+2)^pi//sin^((-1)x)+8a<0 for at least one rea...

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