lab bhattacharjee
Reputation
158,414
99/100 score
 Apr 23 comment 100-th derivative of a function @Joel, Same as $$n=-1$$ right? Apr 23 answered What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$? Apr 23 answered n tends to infinity Apr 23 comment Prove that $6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$ @HPDas, If $T_n=\cos^{2n}x+\sin^{2n}x$ $$T_0=2,T_1=1,T_{n+1}=T_n-\cos x\sin x\cdot T_{n-1}$$ To eliminate $\cos^2x,s=\sin^2x,$ $$6T_5-15T_4=-10T_3+1$$ Apr 22 comment Integral with irrational functions and polynomials Use partial fraction decomposition, $$\dfrac y{1-y^3}=\dfrac A{1-y}+\dfrac{By+C}{1+y+y^2}$$ $$\implies y=A(1+y+y^2)+(1-y)(By+C)$$ Or $y^3-1=(y-1)(y-\omega)(y-\omega^2)$ $y=1\implies3A=1$ Constant $\implies0=A+C\iff C=-A$ Coefficients of $x^2\implies 0=A-B\iff A=B$ Apr 22 comment Prove that $\tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + … + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$ Apr 22 comment Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$ @DavidH, Sorry for the typo Apr 22 revised Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$ edited body Apr 22 answered Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$ Apr 21 comment Show whether this trigonometry series converges $$\le\dfrac{3n}{3^n}$$ Apr 21 awarded Good Answer Apr 21 comment Catalan numbers formula derivation Apr 21 comment 100-th derivative of a function @Stephan. The real part of $e^{i(n\pi/4+x)}$ is $\cos(n\pi/4+x)$ So, the real part of $2^{n/2}e^x\cdot e^{i(n\pi/4+x)}$ is $2^{n/2}e^x\cos(n\pi/4+x)$ which is the required answer Apr 21 awarded Nice Answer Apr 21 comment a/b + b/a is an integer if and only if a = b @Hilly, what are the possible values of $k$ ? Apr 21 comment a/b + b/a is an integer if and only if a = b Observe that $k\pm r$ have the same parity Apr 21 answered a/b + b/a is an integer if and only if a = b Apr 21 comment Optimization of area of rectangle within semicircle Apr 21 comment Is the following solution correct? If $x\ne\pm3i,$ we need $$x^2+1=x^2+9$$ which does not have a finite root Apr 21 answered How to sum binomial coefficients which are multiples of 3?