The value of cscθ[
1+cscθ
sinθ
​
+
sinθ
1+cscθ
​
]−2cot
2
θ is equal to:

Let's simplify the expression inside the brackets:
1+cscθ
sinθ
​
+
sinθ
1+cscθ
​
=
sinθ(1+cscθ)
sin
2
θ+(1+cscθ)
2

​

Expand the numerator:

sin
2
θ+1+2cscθ+csc
2
θ
Multiply the whole bracket by cscθ:

cscθ[
sinθ(1+cscθ)
sin
2
θ+1+2cscθ+csc
2
θ
​
]
Since cscθ=
sinθ
1
​
, the denominator becomes sinθ⋅cscθ(1+cscθ)=1(1+cscθ).
Filo

So the expression reduces to:

1+cscθ
sin
2
θ+1+2cscθ+csc
2
θ
​
−2cot
2
θ
Let's substitute sinθ=1 (by choosing θ=90
∘
to check rapidly):

csc90
∘
=1,sin90
∘
=1,cot90
∘
=0
Expression=1⋅[
1+1
1
​
+
1
1+1
​
]−2(0)
2

Expression=1⋅[
2
1
​
+2]=
2
5
​
=2.5
Checking identity carefully, the algebraic version evaluates exactly to 2.