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The least value of(further question is present in image)A 3B 5C 9D 2

Sudhanva G V , 6 Years ago
Grade 11
anser 1 Answers
Devang

Last Activity: 6 Years ago

(secA  +  cosA)2  +  (cosecA + sinA)2  = sec2A + 2*secA*cosA + cos2A +
                                                                                 cosec2A  +2*cosecA*sinA +   sin2A
                                                            = sec2A  + cosec2A  +2 +2 +1    (  secA*cosA  = 1 
                                                                                                                  cosecA*sinA  = 1
                                                                                                                  sin2A + cos2A = 1  )
                                                             =  1/cos2A  +  1/sin2A  +5
                                                             = sin2A  +  cos2A /sin2A * cos2A  +5
                                                             =  1/ sin2A * cos2A  + 5
                                                             =   4/4sin2A * cos2A   +5
(secA  +  cosA)2  +  (cosecA + sinA)2  = 4/sin22A   +5               ….......(1)
we have,
               -1 \leqslant  sin 2A  \leqslant 1
              0  \leqslant   sin22A  \leqslant 1      (do squre on both side)
              \infty  \geqslant  1/ sin22A  \geqslant 1
              \infty  \geqslant  4/sin22A  \geqslant  4  (multiply bu 4)
              \infty  \geqslant   4/sin22A +5  \geqslant  9   (add 5)    …....(2)
 
on eq. (1) & (2) 
least value of  (secA  +  cosA)2  +  (cosecA + sinA)2 = 9

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