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We have 2tan(th) / (1 + tan^2(th)) = sin(2th)

We know that tan(A) = sin(A) / cos(A), and 1 + tan^2(A) = sec^2(A)

Therefore => (2sin(th) / cos(th)) / sec^2(th)

=> 2sin(th)*cos^2(...

We have 3^(2x+1) = 4^100

=> log(3^(2x+1)) = log(4^100)

=> (2x+1)log(3) = 100log(4)

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We want to rearrange the expression to the form (1+y)^n so we can use the general result: (1+y)^n=1+ny+[n(n-1)/2]y^2+[n(n-1)(n-2)/3!]y^3+... 1/(2+5x)^3 = (2+5x)^-3 = [2(1+5x/2)]^-3 = (2^-3)(1+5x/2)^-3 usi...

Amount of paint needed:

White paint = (18 ÷ 5) × 3 = 10.8 litres

Blue paint = (18 ÷ 5) × 2 = 7.2 litres 

Cost of white paint = 10.8 x £2.80 = £30.24

Cost of blue paint = 7.2 x ...