# Show that: cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65) - Mathematics and Statistics

Sum

Show that:

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

#### Solution

Let a = "cos"^-1 (4/5) and b = "cos"^-1 (12/13)

Let a = "cos"^-1 (4/5)

cos a = 4/5

We know that

sin2a = 1 - cos2

sin a = sqrt (1-"cos"^2 "a")

= sqrt (1 - (4/5)^2) = sqrt (1 - 16/25)

= sqrt ((25-16)/25) = sqrt (9/25) = 3/5

Let b = "cos"^-1 (12/13)

cos b = 12/13

W know that

sin2b = 1 - cos2

sin b = sqrt (1 - "cos"^2 "b")

= sqrt (1 - (12/13)^2) = sqrt (1 - 144/169)

= sqrt ((169-144)/169) = sqrt (25/169) = 5/13

We know that

cos (a+b) = cos a cos b - sin a sin b

Putting values

cos a = 4/5 , sin a = 3/5

& cos b = 12/13 , sin b = 5/13

cos (a+b) = 4/5 xx 12/13 xx 3/5 xx 5/13

= 48/65 - 3/13

= (48 - 15)/65

= 33/65

∴ cos (a+b) = 33/65

a + b = cos-1 (33/65)

"cos"^-1 4/5 + "cos"^-1 (12/15) = "cos"^-1 (33/65)

Hence LH.S = R.H.S

Hence proved.

Concept: Basic Concepts of Trigonometric Functions
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