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About Vijilant
Expertise
Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.
Experience
Teacher of math for 50 years
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ATL
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Journal of mathematics and its applications
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BSc Hons Liverpool
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State Scholarship 1955
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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things.
One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.
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You are here: Experts > Science > Mathematics > Number Theory > trigonometry
Expert: Vijilant - 10/31/2009
Question
Sir, please solve these problems soon,
If cos2B=cos(A+C)/cos(A-C) then show that tanA,tanB and tanC are in G.P.
Prove that (1/cos0°+cos1°)+(1/cos1° +cos2°)+.......+ (1/cos88° +cos90°)= cos1°/sin^2 1°
Find all x between -pi/2 and pi/2 such that 1-sin^4x-cos^2x=1/16
Evaluate sin{n.pi+(-1)^n.(pi/4)}, where n is an integer
If 3tanΘtanØ1 then prove that 2cos(Θ+Ø)=cos(Θ-Ø)
Thanks a lot!
Answer
Hello Sneha
This is not my expertise, but I will help you with some of them.
cos(2B) = 2 cos^2(B) - 1 = 2/sec^2(B) -1 = (1-tan^2(B))/(1+tan^2(B))
Then, clearing the fraction:
cos(A-C) -cos(A-C)tan^2(B) = cos(A+C) +cos(A+C)tan^2(B)
tan^2(B)(cos(A+C)+cos(A-C)) = cos(A-C)-cos(A+C)
tan^2(B)*2cos(A)cos(C) = 2sin(A)sin(C)
tan^2(B) = tan(A)tan(C) provingthey are in GP.
In the third one, use 1-cos^2(x) = sin^2(x) and you have a quadratic in sin^2(x).
Solve this to give sin^2(x) = (2+/- 3^(1/2))/4.
The rest is easy.
The fourth one. Consider odd and even n separately and show that in both cases you get 2^(-1/2).
The other 2 contain misprints.
Best wishes
vijilant
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