Question

Упростите выражение
cos²(α-π/6)+cos²(α+π/6)+sin²α

Answer 1

cos²(α-π/6)+cos²(α+π/6)+sin²α=
=(cosαcosπ/6+sinαsinπ/6)²+(cosαcosπ/6-sinαsinπ/6)²+sin²α=
=cos²αcos²π/6+2cosαcosπ/6sinαsinπ/6+sin²αsin²π/6+
+cos²αcos²π/6-2cosαcosπ/6sinαsinπ/6+sin²αsin²π/6+sin²α=
=2cos²αcos²π/6+2sin²αsin²π/6+sin²α=
=2·3/4cos²α+2·1/4sin²α+sin²α=
=3/2cos²α+3/2sin²α=3/2(cos²α+sin²α)=3/2;