Question

If x= cos^2 tetha /(1-cos^2teta) + 2csc^2teta and y = 2cot^2 teta, express y in terms of x

Answer 1

We'll use the fundamental trigonometric identities:

$sin^2 \theta +cos^2 \theta =1, \textrm{hence } 1-cos^2 \theta = sin^2 \theta.$

$1+cot^2 \theta = csc^2 \theta,$

and $cot \theta =\frac{cos \theta}{sin \theta}.$

Now we can modify the expression for x:

$x= \frac{cos^2 \theta }{1-cos^2\theta} + 2csc^2\theta=\frac{cos^2 \theta }{sin^2\theta} + 2csc^2\theta= cot^2 \theta+2(1+cot^2 \theta)=$

$=cot^2 \theta+2+2cot^2 \theta=3 cot^2 \theta +2=\frac{3}{2}(2cot^2 \theta)+2=\frac{3}{2}y+2.$

Hence

$x=\frac{3}{2}y+2,$

$2x=3y+4,$

$3y=2x-4,$

$y=\frac{2x-4}{3}.$

Answer. $y=\frac{2x-4}{3}.$