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show that 2 sinx+ 2 cosx >2 1 -1/square root 2 2)if tana and tan b are theroots of equation x 2 +px+qthen1) sin (a+b)=-p 2)tan(a+b)=p/q-1 3)cos a+b)1-q

show that 2sinx+2cosx >21 -1/square root 2
2)if tana and tan b are theroots of equation x2+px+qthen1) sin (a+b)=-p 2)tan(a+b)=p/q-1  3)cos a+b)1-q

Grade:12th pass

2 Answers

Vikas TU
14149 Points
5 years ago
U must know the range of sinx and cosx.
similarly               sinx lessthan equal to 2 and greater than equal to ½.................(1)
            1/2 > = 2^cosx > = 2.............(2)
remember cosx is decreasing function hence inequality would change.
add both the eqns. we get,
 0.5 + 2> =  2^cosx + 2^sinx > = 0.5 + 2
 
.ie.  2^cosx + 2^sinx subtends at value 2+ 0.5  = 3.5 > 21 -1/square root 2
H.P.
Madhav Karnani
34 Points
4 years ago
Use A.M. >= G.M. for positive real numbers.2^(sin x) and 2^(cos x) are positive real numbers for all x € R.So, {2^(sin x) + 2^(cos x)}/2 >= sqrt [2^(sin x) × 2^(cos x)] ......... (1)sqrt [2^(sin x) × 2^(cos x)] = 2^{1/2 × (sin (x) + cos (x))}Now, 1/2 × (sin (x) + cos (x)) = 1/sqrt (2) * [ 1/sqrt (2)* sin (x) + 1/sqrt (2)*cos (x)] = 1/sqrt (2) *[ cos (pi/4 - x) ]Now, 2^(sin x) + 2^(cos x) >= 2^[{1/sqrt (2) *cos (pi/4 - x) } +1] ............ (2)***(MULTIPLYING 2 TO BOTH SIDES)Now, -1

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