PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions

`tan^-1  1/4+tan^-1  2/9=1/2cos^-1  3/2=1/2sin^-1(4/5)`

`tan^-1  2/3=1/2tan^-1  12/5`

`tan^-1  1/7+2tan^-1  1/3=pi/4`

`sin^-1  4/5+2tan^-1  1/3=pi/2`

`2sin^-1  3/5-tan^-1  17/31=pi/4`

`2tan^-1  1/5+tan^-1  1/8=tan^-1  4/7`

`2tan^-1  3/4-tan^-1  17/31=pi/4`

`2tan^-1(1/2)+tan^-1(1/7)=tan^-1(31/17)`

`4tan^-1  1/5-tan^-1  1/239=pi/4`

If `sin^-1  (2a)/(1+a^2)-cos^-1  (1-b^2)/(1+b^2)=tan^-1  (2x)/(1-x^2)`, then prove that `x=(a-b)/(1+ab)`

Prove that

`tan^-1((1-x^2)/(2x))+cot^-1((1-x^2)/(2x))=pi/2`

Prove that

`sin{tan^-1  (1-x^2)/(2x)+cos^-1  (1-x^2)/(2x)}=1`

If `sin^-1  (2a)/(1+a^2)+sin^-1  (2b)/(1+b^2)=2tan^-1x,` Prove that  `x=(a+b)/(1-ab).`

Show that `2tan^-1x+sin^-1  (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.

Find the value of the following:

`tan^-1{2cos(2sin^-1  1/2)}`

Find the value of the following:

`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1

Solve the following equation for x:

`tan^-1  1/4+2tan^-1  1/5+tan^-1  1/6+tan^-1  1/x=pi/4`

Solve the following equation for x:

`3sin^-1  (2x)/(1+x^2)-4cos^-1  (1-x^2)/(1+x^2)+2tan^-1  (2x)/(1-x^2)=pi/3`

Solve the following equation for x:

`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`

Solve the following equation for x:

`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`