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Prove `Cos^(-1) 4/5 + Cos^(-1) 12/13 = Cos^(-1) 33/65` - CBSE (Science) Class 12 - Mathematics

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Question

Prove `cos^(-1)  4/5 + cos^(-1)  12/13 = cos^(-1)  33/65`

Solution

Let `cos^(-1)  4/5 = x`. Then, `cos x = 4/5 => sin x = sqrt (1 - (4/5)^2) = 3/5`

`:. tan x =  3/4 => x =  tan^(-1)  3/4`

`:. cos^(-1)  4/5 -  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 `cos^(-1)  33/65 = z`. Then `cos z = 33/65 => sin z = 56/65`

`:. tan z = 56/33 => z = tan^(-1)  56/33`

`:. cos^(-1)  33/65 = tan^(-1)  56/33`  ....(3)

Now, we will prove that:

L.H.S = `cos^(-1)  4/5 + cos^(-1)  12/13`

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

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

`= tan^(-1)  (36+20)/(48-15)`

`= tan^(-1)  56/33`

`= tan^(-1)  56/33`    [by(3)]

= R.H.S

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APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 2: Inverse Trigonometric Functions
Q: 5 | Page no. 51
Solution Prove `Cos^(-1) 4/5 + Cos^(-1) 12/13 = Cos^(-1) 33/65` Concept: Properties of Inverse Trigonometric Functions.
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