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This is question 20, Exercise 6C. "Find the equations of the four tangents common to the hyperbola x^2 - 2y^2 = 4 and the circle x^2 + y^2 = 1. Find the points of contact of these tangents with the circle." [Hint: Let xx1 + yy1 = 1 be tangent to x^2 + y^2 = 1 at P(x1, y1)] Here's what I tried...

Hey, I just can't seem to prove this identity: [1 + cosec^2 (A) tan^2 (C)] [1 + cot^2 (A) sin^2 (C)] ______________________ = ___________________ [1 + cosec^2 (B) tan^2 (C)] [1 + cot^2 (B) sin^2 (C)] It'll be of great help if someone could solve this. Thanks.

I'm stuck on question 28 Exercise 5E on the Patel 4 Unit book. "Show that the polynomial x^n +mx - b = 0 has a multiple root provided (m/n)^n + (b/(n-1))^(n-1) = 0. Find this root." I've tried solving it simultaneously but didn't arrive at the relation stated by the book. Any help...

Suppose that z^7 = 1 where z=/=1 (i) Deduce that z^3 + z^2 + z + 1 + 1/z + 1/z^2 + 1/z^3 = 0 (ii) By letting x = z + 1/z reduce the equation in (i) to a cubic equation in x. (iii) Hence deduce that (cos pi/7)(cos 2pi/7)(cos 3pi/7) = 1/8 I got (i) and (ii) but have no idea on how to...