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Grade: 12th pass

                        

If 0 ≤ x ≤ π/2 , then show that cos(sinx) > sin(cosx).

2 years ago

Answers : (1)

Arun
24742 Points
							
Dear Pawan
 
We have to prove : 

cos (sinx) - sin (cosx) > 0 

=> cos (sinx) - cos ( π/2 - cosx ) > 0 

=> 2 sin [ (π/4) + (1/2). (sinx - cosx) ].sin [ (π/4) - (1/2). (sinx - cosx) ] > 0 ...........(1) 

If we could prove that both the factors on the left hand side of (1) are positive then the result obtained above (1) is proved. 

Since I sinx - cosx I = I √2 sin (x-π/4) I ≤ √2

We have - π/2

=> - π/4

So that 0

And therefore sin [ (π/4) + (1/2). (sinx - cosx) ]. > 0 ie Positive. 

Similarly we can prove that 

sin [ (π/4) - (1/2). (sinx - cosx) ] > 0 ie Positive 

Hence (1) is true. ......................... Proved. 
2 years ago
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