Question

Simplify (1 - cos theeta ) ( 1 + cos theeta ) ( 1 + cot² theeta​

Answer 1

Answer :-

Consider,

(1 - cosø) (1+cosø) (1+ cot²ø)

(a-b)(a+b) = a²-b²

=> ( 1 - cos²ø) ( 1+cos²ø)

=> (sin²ø)(cosec²ø)

=> sin²ø - 1/sin²ø

Cancelling sin²ø ,

=> 1

Answer 2

Answer:

$\left(1-\cos \left(\theta\right)\right)\left(1+\cos \left(\theta\right)\right)\left(1+\cot ^2\left(\theta\right)\right)=1$

Step-by-step explanation:

$\left(1-\cos \left(\theta\right)\right)\left(1+\cos \left(\theta\right)\right)\left(1+\cot ^2\left(\theta\right)\right)$

$\gray{\mathrm{Use\:the\:following\:identity}:\quad \:-\cot ^2\left(x\right)+\csc ^2\left(x\right)=1}$

$\gray{\mathrm{Therefore\:}1+\cot ^2\left(x\right)=\csc ^2\left(x\right)}$

$=\left(1+\cos \left(\theta\right)\right)\csc ^2\left(\theta\right)\left(1-\cos \left(\theta\right)\right)$

$\gray{\mathrm{Expand}\:\left(1-\cos \left(x\right)\right)\left(1+\cos \left(x\right)\right)=1-\cos ^2\left(x\right)}$

$\gray{\mathrm{Use\:the\:following\:identity:}\:1-\cos ^2\left(x\right)=\sin ^2\left(x\right)}$

$=\csc ^2\left(\theta\right)\sin ^2\left(\theta\right)$

$\gray{\mathrm{Use\:the\:following\:identity}:\quad \csc \left(x\right)=\dfrac{1}{\sin \left(x\right)}}$

$=\left(\dfrac{1}{\sin \left(\theta\right)}\right)^2\sin ^2\left(\theta\right)$

$\gray{\left(\dfrac{1}{\sin \left(\theta\right)}\right)^2=\dfrac{1}{\sin ^2\left(\theta\right)}}$

$=\dfrac{1}{\sin ^2\left(\theta\right)}\sin ^2\left(\theta\right)$

$\gray{\mathrm{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}}$

$=\dfrac{\sin ^2\left(\theta\right)}{\sin ^2\left(\theta\right)}$

$\gray{\mathrm{Cancel\:the\:common\:factor:}\:\sin ^2\left(\theta\right)}$

$=1$