1) sin⁴α + cos⁴α = sin⁴α - 2cos²αsin²α + cos⁴α + 2cos²αsin²α = (cos²α - sin²α)² + 2cos²αsin²α = cos²2α + 0,5sin²2α = 0,5(1 + cos4α) + 0,25(1 - cos4α) = 0,5 + 0,5cos4α + 0,25 - 0,25cos4α = 0,75 + 0,25cos4α = 3/4 + (cos4α)/4 = (3 + cos4α)/4.
2) sin2α·sin5α·cos(π/2 - 7α) - cos2α·cos(π/2 + 5α)·cos7α = 1/6;
sin2α·sin5α·sin7α + cos2α·sin5α·cos7α = 1/6;
sin5α(sin2α·sin7α + cos2α·cos7α) = 1/6;
sin5αcos(7α - 2α) = 1/6;
sin5αcos5α = 1/6|·2;
2sin5αcos5α = 1/3;
sin10α = 1/3;
cos20α = 1 - 2sin²10α = 1 - 2·1/9 = (9 - 2)/9 = 7/9
3)