To prove the identity 1/(1 - sin θ) + 1/(1 + sin θ) = 2sec² θ, we combine the fractions on the left and use the Pythagorean identity to show that it simplifies to the right side of the equation.
The student's question deals with verifying a trigonometric identity involving sine and secant functions. To prove the identity 1/(1 - sin θ) + 1/(1 + sin θ) = 2sec² θ, we must manipulate the left side of the equation so it matches the right side. We start by finding a common denominator for the fractions on the left side:
1/(1 - sin θ) + 1/(1 + sin θ)
= [(1 + sin θ) + (1 - sin θ)] / [(1 - sin θ)(1 + sin θ)]
= (1 + sin θ + 1 - sin θ) / (1 - sin² θ)
= 2 / (1 - sin² θ)
Using the Pythagorean identity, sin² θ + cos² θ = 1, we see that 1 - sin² θ equals to cos² θ. Therefore, we can rewrite the fraction:
= 2 / cos² θ
= 2sec² θ
Thus, the original identity is proven to be true.