Question

Cos square 15-cos square 75=root3/2

Answer 1

Explanation:

Show that cos^2 (45°)-sin^2 (15°) = √3/4?

cos245∘−sin215∘cos2⁡45∘−sin2⁡15∘

=cos245∘−sin2(45∘−30∘)(LHS)(LHS)=cos2⁡45∘−sin2⁡(45∘−30∘)

We know that sin(A−B)=sinA⋅cosB−cosA⋅sinBsin⁡(A−B)=sinA⋅cosB−cosA⋅sinB

Therefore, LHS =cos245∘−(sin45∘⋅cos30∘−cos45∘⋅sin30∘)2=cos2⁡45∘−(sin⁡45∘⋅cos⁡30∘−cos⁡45∘⋅sin⁡30∘)2

=12−(12–√×3–√2−12–√×12)2=12−(12×32−12×12)2

=12−(3–√22–√−122–√)2=12−(322−122)2

=12−(3–√−122–√)=12−(3−122)

== 12−((3–√−1)28)12−((3−1)28)

=12−(3–√−1)28=12−(3−1)28

=12−4−23–√8=12−4−238

=12−2−3–√4=12−2−34

=2−(2+3–√)4=2−(2+3)4

=3–√4==34=RHS.

Hence, the problem is solved.