Edexcel C3 June 2015 Q1: tan(x)=p, where p is a constant. Using standard trigonometric identities, find the following in terms of p. a) tan(2x). b) cos(x). c) cot(x-45).

a) tan(A+B)=(tanA+tanB)/(1-tanAtanB) So, tan(2x)=[tan(x)+tan(x)]/[1-(tanx)(tanx)]. Therefore, tan(2x)=[2tan(x)]/[1-tan^2(x)] = 2p/(1-p^2). b) cos(x)=1/sec(x). Using other trigonometric identities, we know that sec^2(x)=1+tan^2(x). Hence, cos(x)=1/sqrt[1+tan^2(x)] = 1/sqrt(1+p^2). c) cot(x-45)=1/tan(x-45). tan(x-45)=[tan(x)-tan(45)]/[1+tan(x)tan(45)] tan(x-45)=[tan(x)-1]/[1+tan(x)] Therefore, cot(x-45)=[1+tan(x)]/[tan(x)-1]=(1+p)/(p-1)