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