# Prove that : cos^-1 (12/13)  + sin^-1(3/5) = sin^-1(56/65) - Mathematics

Sum

Prove that :

cos^-1 (12/13)  + sin^-1(3/5) = sin^-1(56/65)

#### Solution

Let sin^-1  3/5 = "x"
Then, sin"x" = 3/5 ⇒ cos"x" = sqrt(1 - (3/5)^2) = sqrt(16/25) = 4/5

∴ tan"x" = 3/4 ⇒ "x" = tan^-1  3/4

∴ sin ^-1  3/5 = tan^-1  3/4 ......(1)

Now,let cos^-1  12/13 = "y".
Then, cos"y" = 12/13 ⇒ sin"y" = 5/13

∴ tan"y" = 5/12 ⇒ "y" = tan^-1  5/12

∴cos^-1  12/13 = tan^-1  5/12  ........(2)

Let sin^-1  56/65 = "z".
Then, sin"z" = 56/65 ⇒ cos"z" = 33/65

∴ tan"z" = 56/33 ⇒ "z" = tan^-1  56/33

∴sin^-1  56/65 = tan^-1  56/33   .........(3)

Now, we have:

L.H.S. = cos^-1  12/13 + sin^-1  3/5

= tan^-1  5/12 + tan^-1  3/4  .........[Ueing (1) and (2)]

=tan^-1  (5/12+3/4)/(1-5/12 . 3/4)  ........[tan^-1"x"+ tan^-1"y"=tan^-1  "x+y"/(1-"xy")]

 = tan^-1  (20+36)/(48-15)

= tan^-1  56/33

= sin^-1  56/65 = "R.H.S"  .........[Using (3)]

Concept: Proof Derivative X^n Sin Cos Tan
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