To find minimum value of 2 csc &#x2061;<!-- ⁡ --> ( 2 x ) + sec &#x20

raulgallerjv 2022-05-23 Answered
To find minimum value of 2 csc ( 2 x ) + sec ( x ) + csc ( x )
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Answers (1)

tatoas9f
Answered 2022-05-24 Author has 11 answers
Method #1:
If we set sin x + cos x = u , u 2 , u 2 = 1 + 2 sin x cos x
we need to minimize
2 u + 1 u 2 1 = 2 u 1
i.e., to maximize u−1
Method #2:
As ( sin x + cos x ) 2 1 2 = 2 sin x cos x
and sin x + cos x = 2 sin ( x + π 4 )
sin x + cos x + 1 sin x cos x = 2 sin x + cos x 1 = 2 2 sin ( x + π 4 ) 1
Now for x ( 0 , π 2 ) ,,
1 2 < sin ( x + π 4 ) 1
Method #3:
2 sin x cos x = cos 2 ( π 4 + x ) = ( 2 sin ( x + π 4 ) ) 2 1 2
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