Question

show that cos^2 45-sin^2 15=√3/4.​

Answer 1

L.H.S.=Cos^2(45°) - Sin^2(15°)

={Cos(45°)}^2 - [Sin (45°-30°)]^2

={Cos(45°)}^2 - [Sin45°×Cos30°-Cos45°×Sin30°]^2

=(1/√2)^2 - [(1/√2 ×√3/2) - (1/√2×1/2)]^2

=1/2 - [√3/2√2 - 1/2√2]^2

=1/2 - (√3–1/2√2)^2

=1/2 - [(√3–1)^2/4×2]

=1/2 - [(√3)^2-2×√3×1+(1)^2/8]

=1/2 - [3–2√3+1/8]

=1/2 - 4+2√3/8

=4–4+2√3/8

=2√3/8

=√3/4

= R.H.S

Hence proved