The value of
sec
2
θ
32
​
−
1+tan
2
θ
20
​
+12sin
2
θ is:

Let's simplify the expression using standard trigonometric identities:
We know that:

sec
2
θ
1
​
=cos
2
θ

1+tan
2
θ=sec
2
θ, so
1+tan
2
θ
1
​
=
sec
2
θ
1
​
=cos
2
θ

Substitute these into the given expression:

Expression=32cos
2
θ−20cos
2
θ+12sin
2
θ
Combine the cos
2
θ terms:

Expression=(32−20)cos
2
θ+12sin
2
θ
Expression=12cos
2
θ+12sin
2
θ
Factor out 12:

Expression=12(cos
2
θ+sin
2
θ)
Using the basic Pythagorean identity sin
2
θ+cos
2
θ=1:

Expression=12×1=12