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Prove the identity: (sinx - tanx)(cosx - cotx) = (sinx - 1)(cosx - 1)

(sinx - tanx)(cosx - cotx)= (sinx - sinx/cosx)(cosx - cosx/sinx)= sinxcosx - (sinx + cosx) + 1= (sinx - 1)(cosx - 1)

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For what values of k does the line y=kx-1 have two distinct points of intersection with the circle (x-2)^2+(y-3)^2=2?

sub y=kx-1 into circle equation, get (k^2+1)x^2-(8k+4)x+18=0for 2 distinct solutions need b^2-4ac>0, ie -8k^2+64k-56>0iff k^2-8k+7<0complete the square: intersections of equation in k satisfy(k-4...

Answered by Christopher B. • Maths tutor

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Prove by induction that the nth triangle number is given by n(n+1)/2

base case: (1 x 2)/2 = 1 as required inductive step: assuming statement holds for n=k, the (k+1)th triangle number is given by k(k+1)/2 + (k+1) by definition=(k^2+3k+2)/2=(k+1)(k+2)/2=(k+1)((k+1)+1)/2resu...

Answered by Christopher B. • Maths tutor

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How would I go about drawing the graph of f(x) = sin(x)/(e^x) for -π≤x≤2π?

Drawing this graph uses the same techniques as with most other functions. Just from inspection, we can see that as x increases, e^x also increases, and so the amplitude of the function will decrease (and ...

Answered by Jonah Z. • Maths tutor

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2 log(x + a) = log(16a^6) where a is a positive constant. How do I find x in terms of a?

The left hand side of the equation can be rewritten as log(x+a)^2, using the power rule for logs. We can then simplify log(x+a)^2=log(16a^6) to (x+a)^2=16a^6. Taking the square root of both sides, we find...

Answered by Emily V. • Maths tutor

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