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Trignometry

Blazing goIITian

Joined:	27 Dec 2010
Post:	1474

8 Feb 2012 12:08:28 IST

3 People liked this

216

this one is good.

Mathematics

suppose A, B and C are the angles of a triangle.

if x some other angle such that

(sinx)^3 = sin(A-x)sin(B-x)sin(C-x)

then prove that cotx = cotA + cotB + cotC.

my solution is longer than two pages (if i don't use cyclic summations).

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jagdish singh

Blazing goIITian

Joined: 19 Jan 2008

Posts: 812

10 Feb 2012 20:04:08 IST

Like 3 people liked this

$\hspace{-16}\mathbf{\cot x=\cot A+\cot B+\cot C}$\\\\\\ $\mathbf{\cot x-\cot A=\cot B+\cot C}$\\\\\\ $\mathbf{\frac{\cos x}{\sin x}-\frac{\cos A}{\sin A}=\frac{\cos B}{\sin B}+\frac{cos C}{\sin C}}$\\\\\\ $\mathbf{\frac{\sin A.\cos x-\cos A.\sin x}{\sin x.\sin A}=\frac{\sin B\cos C+\cos B.\sin C}{\sin B.\sin C}}$\\\\\\ $\mathbf{\frac{\sin (A-x)}{\sin x.\sin A}=\frac{\sin (B+C)}{\sin B.\sin C}=\frac{\sin A}{\sin B.\sin C}}$\\\\\\ $\mathbf{\sin (A-x)=\frac{\sin^2 A.\sin x}{\sin B.\sin C}}$\\\\\\ Similarly $\mathbf{\sin (B-x)=\frac{\sin^2 B.\sin x}{\sin C.\sin A}}$\\\\\\ Similarly $\mathbf{\sin (C-x)=\frac{\sin^2 C.\sin x}{\sin A.\sin B}}$\\\\\\ So $\mathbf{\sin (A-x).\sin (B-x).\sin (C-x) = \sin ^3 x}$$

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