laplace transform using convolution intergral

Question

struggling with the following question:

Evaluate the following convolution integral

cos(x)*cos(x)

any help would be much appreciated, the hint of using the cos trig identity has been given.

Answer 1

The hint of using trig identities is probably as follows $$\cos x*\cos x=\int_{\text{limits}^*}\cos (u')\cdot\cos(u-u')\,du'\\=\int_{\text{limits}}\cos u'(\cos u'\cos u+\sin u'\sin u)\,du'\\=\cos u\int_{\text{limits}}\cos^2u'\,du'+\sin u\int_{\text{limits}}\sin u'\cos u'\,du'$$ which can both be evaluated using identities relating to $\cos2u'$ and $\sin2u'$.

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$*$ - I have written $\text{limits}$ since I'm not sure what regions you are defining $\cos u'$ on, so I suppose you can put this in.