Grade 12th passTrigonometry

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

Pawan joshi

7 Years agoGrade 12th pass

1 Answer

Arun

7 Years ago

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.