Question

simplify (1-cos theta)(1+cos theta)(1+cot squared theta)

Answer 1

1...sin square theta × cosec square theta

Answer 2

Answer:  The required simplified expression is 1.

Step-by-step explanation:  We are given to simplify the following trigonometric expression :

$E=(1-\cos\theta)(1+\cos\theta)(1+\cot^2\theta).$

We will be using the following formulas :

$(i)~1-\cos^2\theta=\sin^2\theta,\\\\(ii)~\cot\theta=\dfrac{\cos\theta}{\sin\theta}\\\\(iii)~(a-b)(a+b)=a^2-b^2.$

The simplification of the given expression is as follows :

$E\\\\=(1-\cos\theta)(1+\cos\theta)(1+\cot^2\theta)\\\\=(1-\cos^2\theta)\left(1+\left(\dfrac{\cos\theta}{\sin\theta}\right)^2\right)\\\\\\=\sin^2\theta\left(1+\dfrac{\cos^2\theta}{\sin^2\theta}\right)\\\\\\=\sin^2\theta\times\dfrac{\sin^2\theta+\cos^2\theta}{\sin^2\theta}\\\\\\=\dfrac{\sin^2\theta}{\sin^2\theta}\times1~~~~~~~~~~~~~~~~~~~[\textup{since }\sin^2\theta+\cos^2\theta=1]\\\\=1.$

Thus, the required simplified expression is 1.