CBSE Class 11-science Answered

given

m²+m' ²+2mm'cosø=1 ....................1

n² +n' ²+ +2nn'cosø=1 ..........................2

and mn + m'n'+(mn'+m'n)cosø=0...............3

tp....... m² +n² =cosec²ø.

from eq 1

(m' + mcosø)² + m² -m²cos²ø = 1

(m' + mcosø)² =1 - m² sin²ø.....................4

similarly

(n' + ncosø)² = 1 - n² sin²ø......................5

now , (m' + mcosø)(n' + ncosø) = m' n'  + (mn'  + m' n )cosø + mncos²ø

                                                          = - mn + mncos²ø    (from 3)

                                                           = -mn(1 -  cos²ø ) = -mnsin²ø

squ both side

(m' + mcosø)² (n' + ncosø)² =  m² n² sin⁴ø.............6

from eq4,5,&6

(1 -  m² sin²ø)(1 -  n² sin²ø ) = m² n² sin⁴ø

(m²  + n² )sin²ø = 1

or   m² +n² =cosec²ø.