Question

cos-1 12/13+2cos-1√64/65+cos-1√49/50=cos-1 1/√2​

Answer 1

•Given:-

Cos-1 12/13+2cos-1√64/65+cos-1√49/50

•To prove :-

Cos-1 12/13+2cos-1√64/65+cos-1√49/50=cos-1 1/√2

•Solution:-

Cos-1(12/13) = tan-1 (5/12)

cos-1(√64/√65) = tan-1 (1/8)

cos-1(√49/√50) = tan-1 (1/7)

so,

Cos-1 12/13+2cos-1√64/65+cos-1√49/50

=tan-1 (5/12)+ 2tan-1 (1/8) + tan-1 (1/7)

=tan-1 (5/12)+ tan-1 (1/8)+ tan-1 (1/8)+tan-1 (1/7)

= tan-1 [5/12+1/8] / [1 - (5/12×1/8)] + tan-1 [1/8+1/7] / [1- (1/8×1/7)]

= tan-1 (4/7) + tan-1 (15/55)

= tan-1 4/7 + tan-1 3/11

= tan-1 [(44+21)/77] /[ 1 - (4×3/77)]

= tan-1 65/65 = tan-1 1 = 45° = cos-1 1/√2