Let cosα = sinβ = 0.8957 and let 0° ≤ β ≤ 90°. if α = 26.4°, what is the value, in degrees, of angle β?

If cosα = sinβ  and 0° ≤ β ≤ 90°, usually it means is a right-triangle, so there's another angle that is not α or β that is 90°, and in which case angles α and β are "complementary angles", namely α + β = 90°, like in the example in the picture below.

what the dickens does all that mean?  well, it means that if  α = 26.4°, then 

α + β = 90°

26.4° + β = 90°

β = 90° - 26.4°.