Question 759298
cot(105°)*tan(15°)
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Identity:{{{tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan (y))}}}
{{{tan(105)=tan(60+45)=(tan(60)+tan(45))/(1-tan(60)tan (45))
=(sqrt(3)+1)/(1-sqrt(3)*1)=(sqrt(3)+1)(sqrt(3)+1)/(1-sqrt(3)(sqrt(3)+1))
=(3+2sqrt(3)+1)/(1-3)=(4+2sqrt(3))/(-2)=-(2+sqrt(3))}}}
cot(105º)=1/tan(105º)=-1/(2+√3)
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Identity:{{{tan(x-y)=(tan(x)-tan(y))/(1+tan(x)tan (y))}}}
{{{tan(15)=tan(60-45)=(tan(60)-tan(45))/(1+tan(60)tan (45))
=(sqrt(3)-1)/(1+sqrt(3)*1)=(sqrt(3)-1)(sqrt(3)-1)/(1+sqrt(3)(sqrt(3)-1))
=(3-2sqrt(3)+1)/(-1+3)=(4-2sqrt(3))/(2)=2-sqrt(3))}}}
tan(15º)≈2-√3
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cot(105°)*tan(15°)={{{(-1/(2+sqrt(3)))(2-sqrt(3))=-(4-4sqrt(3)+3)/(4-3)=-7+4sqrt(3)}}}
Calculator check:
tan(105º)≈-3.73
exact value=-(2+√3)≈3.73
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tan(15º)≈0.2679
exact value=2-√3≈0.2679