Integrating Cosine Squared Using Double Angle Identity

Evaluate $\displaystyle \int \cos^2(\theta) \, d\theta$ using the double angle identity.

When faced with an integral of trigonometric functions like $\int \cos^2(\theta) \, d\theta$, one effective strategy is to use trigonometric identities to simplify the expression before integrating. Specifically, the double angle identity for cosine provides a useful tool in this scenario. The double angle identity for cosine is expressed as $\cos(2\theta) = 2\cos^2(\theta) - 1$, which can be rearranged to express $\cos^2(\theta)$ in terms of $\cos(2\theta)$: $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$. This substitution transforms the original integral into a much more manageable form by turning a quadratic trigonometric expression into a linear one with respect to another angle, which can significantly simplify the integration process.

Once this transformation is made, the integral can be resolved using basic integration rules. Integrating $\frac{1}{2}$ is straightforward, resulting in a linear term. The second part, $\frac{1}{2} \cos(2\theta)$, is also straightforward to integrate since it resembles the derivative format of a sine function through its angle, $2\theta$. This approach showcases the importance of trigonometric identities in integration, serving as a crucial skill in managing complex integral calculations that involve periodic functions like sine and cosine. Understanding and applying these identities not only simplifies the process but also deepens comprehension of the connections between different trigonometric properties and their integrals.

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