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For example, I recently came across the following way to evaluate the integral of $\cos^2 x - \sin^2 x$ without using double angle formulas:

$$\int dx (\cos^2 x - \sin^2 x) = \int dx(\cos x + \sin x)(\cos x - \sin x) = \int u du = \frac{u^2}{2} + C$$

Where $u = \cos x + \sin x$. One can expand the final result to get $\sin x \cos x + C'$ as the final result, i.e. $\frac{1}{2} \sin (2x) + C'$. Though this may take longer, I find this solution valuable because it reminds us that there is more than one solution, even to a seemingly rigid problem like this.

Another example is solving $\lim \limits_{n \to \infty} \sqrt[n]{n}$ using AM-GM and Squeeze theorem:

$\frac{(n - 2)\cdot 1 + 2 \cdot \sqrt{n}}{n} \geq \sqrt[n]{n} \geq 1$ by AM-GM for the first inequality, and by definition for the second

$1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \geq \sqrt[n]{n} \geq 1$ and then squeeze.

I found this solution to be much more interesting than the standard solution which is to take $\ln$ of the expression and find that limit.

Can you show me other examples of this? (I am personally only an advanced high schooler, but I will certainly appreciate answers at any level and hopefully I will be able to fully understand them some time in the future.) Also note that any area of study is acceptable.

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@zyx On the left side, there are $n-2$ additions of $1$ and $2$ additions of $\sqrt{n}$ :) – Michael Tong Mar 18 at 19:04
    
Edited to make it more explicit. Thanks for the explanation. – zyx Mar 19 at 17:08

Perhaps the most standard way to compute the sum $$1+3+5+\ldots +(2n-1)=n^2$$ would be to use the arithmetic series with $a_0=1$, $d=2$.

Here is another solution:

enter image description here

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