# How do you evaluate the definite integral #int (1-sin^2theta)/cos^2theta# from [0,pi/4]?

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To evaluate the definite integral (\int_{0}^{\frac{\pi}{4}} \frac{1 - \sin^2 \theta}{\cos^2 \theta} d\theta), we first simplify the integrand using trigonometric identities.

Given that (\sin^2 \theta + \cos^2 \theta = 1), we can rewrite (\frac{1 - \sin^2 \theta}{\cos^2 \theta}) as (\frac{\cos^2 \theta}{\cos^2 \theta}).

This simplifies further to 1, so the integral becomes (\int_{0}^{\frac{\pi}{4}} 1 d\theta).

Integrating 1 with respect to (\theta) gives (\theta), so the result of the integral is (\left[\theta\right]_{0}^{\frac{\pi}{4}}).

Evaluating this from (0) to (\frac{\pi}{4}) gives (\frac{\pi}{4} - 0), which simplifies to (\frac{\pi}{4}).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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