Let's apply the cos of a sum formula to cos(15∘+15∘), as we know that cos(30∘)=23:
23=cos(30∘)=cos(15∘+15∘)=cos(15∘)×cos(15∘)−sin(15∘)×sin(15∘)
Now by the main trigonometrical identity (sin2α+cos2α=1) we can write :
23=cos2(15∘)−(1−cos2(15∘))=2cos2(15∘)−1
2cos2(15∘)=1+23
cos(15∘)=±21+43
We know also that cos(15∘)>0 (as 0∘<15∘<90∘ ), so we have :
cos(15∘)=21+43=212+3
We can also simplify the last expression by writing :
(x+y)2=2+3
x2+y2=2,xy=23
x2+4x23=2
x4−2x2+43=0
x2=22±4−3
x=21,y=23
So we have :
cos(15∘)=21×(21+23)=42+6
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