We know that Sin^{2}x + Cos^{2}x = 1 for all values of x.
If Sin x or Cos x is equal to –1 or 1, then Sin^{2014}x + Cos^{2014}x will be equal to 1.
Sin x is equal to –1 or 1 when x = –4.5π or –3.5π or –2.5π or –1.5π or –0.5π or 0.5π or 1.5π or 2.5π or 3.5π or 4.5π.
Cosx is equal to –1 or 1 when x = –5π or –4π or –3π or –2π or –π or 0 or π or 2π or 3π or 4π or 5π.
For all other values of x, Sin^{2014} x will be strictly lesser than Sin^{2}x.
For all other values of x, Cos^{2014} x will be strictly lesser than Cos^{2}x.
We know that Sin^{2}x + Cos^{2}x is equal to 1. Hence, Sin^{2014}x + Cos^{2014}x will never be equal to 1 for all other values of x. Thus there are 21 values.
Answer choice (C)
Correct Answer: 21 Values