(sin^2 θ)(csc^2 θ + sec^2 θ) = sec^2 θ
Let's simplify the left side of the equation:
(sin²θ)(csc²θ + sec²θ)
To simplify this expression, we can use the trigonometric identities:
csc² θ = 1/sin² θ
sec² θ = 1/cos² θ
Substituting these identities into the expression, we have:
(sin² θ)(1/sin^2 θ + 1/cos² θ)
Next, we can find a common denominator by multiplying the fractions:
[(sin² θ * cos² θ) + (sin² θ * sin² θ)] / (sin² θ * cos² θ)
Using the identity sin² θ + cos² θ = 1, we can simplify further:
[(sin² θ * cos²θ) + (1 - cos²θ)] / (sin² θ * cos² θ)
Expanding the numerator:
[sin² θ * cos² θ + 1 - cos² θ] / (sin² θ * cos² θ)
Combining like terms:
[sin² θ * cos² θ - cos² θ + 1] / (sin² θ * cos² θ)
Factoring out cos² θ:
[cos² θ (sin² θ - 1) + 1] / (sin² θ * cos² θ)
Using the identity sin² θ - 1 = -cos² θ:
[cos² θ (-cos² θ) + 1] / (sin² θ * cos² θ)
Simplifying the numerator:
[-cos^4 θ + 1] / (sin² θ * cos² θ)
Applying the identity cos² θ = 1 - sin² θ:
[-(1 - sin^² θ)² + 1] / (sin² θ * (1 - sin² θ))
Expanding and simplifying the numerator:
[-(1 - 2sin² θ + sin² θ) + 1] / (sin² θ * (1 - sin² θ))
Distributing the negative sign:
[-1 + 2sin² θ - sin^4 θ + 1] / (sin² θ * (1 - sin² θ))
Canceling out the -1 and +1 terms:
[2sin² θ - sin^4 θ] / (sin² θ * (1 - sin² θ))
Using the identity 1 - sin² θ = cos² θ:
[2sin² θ - sin^4 θ] / (sin² θ * cos² θ)
Rearranging the terms:
[(sin² θ)(2 - sin² θ)] / (sin² θ * cos² θ)
Canceling out the sin² θ terms:
(2 - sin² θ) / cos² θ
Using the identity 2 - sin² θ = 1 + cos² θ:
(1 + cos² θ) / cos² θ
Simplifying further:
1/cos^
2 θ + cos² θ/cos² θ
Which equals:
sec² θ + 1
Therefore, the left side of the equation simplifies to sec^2 θ + 1, not just sec² θ.