Show that 1+cot^2(x)=cosec^2(x). Need to remember that:sin 2 (x)+cos 2 (x)=1 (eq.1)Divide the whole eq.1 by sin 2 (x)to get:sin 2 (x)/sin

Using the identity:cot^2(x) + 1 = cosec^2(x),we see the left-hand side simplifies to:cot^2(x) + 1 = 1 + 2cot(x).Rearranging,cot^2(x) – 2cot(x) = 0.We now ...

Solve the equation 5*cosec(x) + cosec^2(x) = 2 - cot^2(x) in the interval 0<x<2*pi, giving the values of x in radians to three significant figures.

Convert all cosec/cot/sec functions into functions using sin/tan/cos = 1 / (sin2x) + cos(2x) / sin(2x). Combine the two fractions into one.

This question makes good use of the trigonometric identities tan 2 x + 1 = sec 2 x and 1 + cot 2 x = cosec 2 x which can be easily recited in the exam.

By substituting in the identity cot^2(x) = cosec^2(x) - 1 you get the required form. If you can remember this identity it can be derived from sin^2(x) + cos^2(x) ...

Cosec2A - cot2A= tanA. Left hand side=1/sin2A - cos2A/sin2A =(1- cos2A)/sin2A =(1-(1- 2 sin^2⁡ A)/ 2sinAcosA =(1-1 + ( 2 sin^2⁡ A))/ 2sinAcosA =sinA/cosA

By multiplying throughout by cos 2 (x), we get(sec 2 x/tan 2 x)=1/sin 2 (x)=cosec 2 (x)as required.

For this we have to use trignometric identities, e.g Tan(x)= sin(x)/cos(x), sin2(x) + cos2(x) = 1, 1/sin(x) = cosec(x) tan(x) + cot(x) = sin(x)/cos(x) + ...

An example of this would be in remembering the trigonometric identity '1+cot(^2)x=cosec(^2)x', by thinking of it as the phrase 'One in a cot is cosy', where ...