Which one of the two is greater,
sin(cos a) or cos(sin a)?

• We will make use of the following trigonometric identities:
  
  sin A = cos(pi/2 − A)-------------------------------------(1)
  cos A − cos B = −2 sin(1/2(A + B)) sin(1/2(A − B))----------(2)
  A cos x + B sin x = R cos (x − c)-----------------------(3)
  where R = (A2 + B2), and c = arctan(B/A)
  
  Applying the above, we obtain
  
  cos(sin x) − sin(cos x) = cos(sin x) − cos(pi/2 − cos x) from (1).
  = −2 sin[1/2(sin x−cos x +pi/2)]sin[1/2(sin x + cos x − /2)]from (2).
  = −2 sin(−1/2(2)^1/2 cos(x +pi/4)+pi/4)
  X
  sin(1/2(2)^1/2cos(x − pi/4)− /4) from (3).
  
  Since pi/4 > 1/2 (2)^1/2, 0 < −1/2 (2)^1/2cos(x +pi/4) + pi/4 0, for all x.
  
  Similarly, −pi/2 < 1/2 (2)^1/2cos(x −pi/4)− pi/4 < 0, for all x.
  
  Hence sin(1/2 (2)^1/2cos(x −pi/4)−pi/4) < 0, for all x.
  
  It follows that −2 sin(−1/2cos(x + pi/4) +pi/4) sin(1/2cos(x −pi/4)−pi/4) > 0.
  
  Therefore cos(sin x) > sin(cos x) for all real x.
• 12 years agoHelpfull: Yes(8) No(6)
• cos(sin a) > sin(cos a).
  sol:-
  let a=180(degree)
  now,
  cos(sin 180')=cos(0)=1
  
  sin(cos 180')=sin(-1)=-0.84147
• 12 years agoHelpfull: Yes(7) No(0)
• ans cos(sin a) is greater than sin(cos a) .because when sin a=0 then cos(sin a)=1
  and when cos a=0 then also sin(cos a)=0.
• 12 years agoHelpfull: Yes(0) No(0)