Question

Prove that cot7.5 degrees=√ 2-√ 3 -√ 4 +√ 6???Please prove it..

Answer 1

cos15°
=cos(45°-30°)
=cos45°cos30°+sin45°sin30°
=(1/√2)(√3/2)+(1/√2)(1/2)
=(√3+1)/2√2
cot²7.5°
=2cos²7.5°/2sin²7.5°
=(1+cos15°)/(1-cos15°)
=[1+(√3+1)/2√2]/[1-(√3+1)/2√2]
=[(2√2+√3+1)/2√2]/[(2√2-√3-1)/2√2]
=[2√2+(√3+1)]/[2√2-(√3+1)]
=[{2√2+(√3+1)}{2√2+(√3+1)}]/[{2√2-(√3+1)}{2√2+(√3+1)}]
={2√2+(√3+1)}²/{(2√2)²-(√3+1)²}
={8+2×2√2×(√3+1)+(√3+1)²}/{8-(√3+1)²}
={8+4√2(√3+1)+3+2√3+1}/{8-(3+2√3+1)}
=(12+4√6+4√2+2√3)/(8-4-2√3)
=(6+2√6+2√2+√3)/(2-√3)
={(6+2√6+2√2+√3)(2+√3)}/{(2-√3)(2+√3)}
=(12+4√6+4√2+2√3+6√3+6√2+2√6+3)/(4-3)
=15+6√6+10√2+8√3
=5(3+2√2)+2√3(3√2+4)
=5(3+2√2)+2√6(3+2√2)
=(3+2√2)(5+2√6)
∴, cot7.5°=√{(3+2√2)(5+2√6)}
Please check the question. Thank you!