(C3) Show that 4csc^2(x) - cot^2(x) = k can be expressed as sec^2(x) = (k-1)/(k-4) where k != 4
The student can answer this in several ways. One using the simple, known identities csc= 1/sin, cot=1/tan, sec=1/cos, tan=sin/cos, sin^2 + cos^2 = 1 and basic algebra is the following:
4csc^2 - cot^2 = k4/sin^2 - 1/tan^2 = k Substitute inverse fomulae4/sin^2 - cos^2/sin^2 = k Substitute tan4 - cos^2 = ksin^2 4 - cos^2 = k(1-cos^2) Write in terms of Coskcos^2 - cos^2 = k - 4cos^2(k - 1) = k-4 Gather Cos terms(k-1)/(k-4) = sec^2 Write in terms of inverses
FH
Related Maths A Level answers
All answers ▸How to write an algebraic fraction in a given form e.g. (3+13x-6x^2)/(2x-3) as Ax + B + C/(2x-3) where A, B and C are natural numbers
Consider f(x)=x/(x^2+1). Find the derivative f'(x)
The equation x^2 + 3px + p = 0, where p is a non-zero constant, has equal roots. Find the value of p.
Given that y = (3x^4 + x)^5, find dy/dx using the chain rule.