Recent content by DrunkenOldFool

In a triangle $ABC$, the sides opposite to vertices $A,B,C$ are $a,b,c$ respectively. I have to prove $$ \frac{\tan\left( \frac{A}{2}\right)}{(a-b)(a-c)}+\frac{\tan\left( \frac{B}{2}\right)}{(b-c)(b-a)}+\frac{\tan\left( \frac{C}{2}\right)}{(c-a)(c-b)} = \frac{1}{\Delta}$$ $\Delta$ denotes the...

This question was asked in my exam and I could not answer it. I would like to know how it can be solved. If $l$ and $m$ are variable real numbers such that $5l^2+6m^2-4lm+3l=0$, then a variable line $lx+my=1$ always touches a fixed parabola, whose axis is parallel to the x-axis. (a) Find the...

$a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that $$ab^2c^3 \leq \frac{1}{432}$$

Thank You!

Solve for complex number, $z$:\[\text{arg}\left( \frac{3z-6-3i}{2z-8-6i}\right)=\frac{\pi}{4}\]and \[|z-3+i|=3\]The problem I am facing is that when I substitute $z=x+iy$, the equations become extremely complicated. There has to be another tricky method which I am not able to figure out.

Tangents are drawn to the circle $x^2+y^2=32$ from a point $A$ lying on the x-axis. The tangents cut the y-axis at a point $B$ and $C$, then find the coordinate(s) of $A$ such that the area of $\Delta ABC$ is minimum.

Thank You! This was extremely helpful.

Find the equations of the sides of square inscribed in the circle $3(x^2+y^2)=4$, one of whose sides is parallel to the line $x-y=7$.

Thank you very much, sir!

Let $f(x)=x^{13}+2x^{12}+3x^{11}+\cdots +13x+14$ and $\alpha=\cos\frac{2\pi}{15}+i\sin\frac{2\pi}{15}$. Find the value of $f(\alpha)f(\alpha ^2) \cdots f(\alpha ^{14})$. (the answer given in my book is 15^(13)). Please help me! I beg of you...

If $\cos \alpha +\cos \beta + \cos \gamma=0$ and $\cos 3 \alpha +\cos 3\beta +\cos 3\gamma = \lambda \cos \alpha \cos \beta \cos \gamma$. What is the value of $\lambda$?

If $\sin x +\cos y=a$ and $\cos x+\sin y =b $, then what is $\tan\dfrac{x-y}{2}$ in terms of $a$ and $b$?

Thank You Sudharaka! Can you suggest any other simpler method?(f)

If $\alpha>0$, $\beta< \pi$ and $\cos(\alpha)+\cos(\beta)-\cos(\alpha+\beta)=3/2$, then what is the value of $\sin(\alpha)+\cos(\beta)$?

Thank You!:)