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The question in the picture is a relatively easy one to answer. But while learning Trigonometry, I had to answer countless questions asking to prove certain relations between obscure and complicated expressions. To give you another example: If sin(y+z-x), sin(z+x-y), sin(x+y-z) are in arithmetic progression, prove that tan x, tan y and tan z are also in arithmetic progression.
Can someone explain to me what use they serve practically?
I suspect the intended use of these questions was to provide practise using trigonometric identities and also the idea of arithmetic progression (which I think would both appear in the UK A-level syllabus, or maybe GCSE, I can't remember). Memorising formulae is not much fun, and having done so is not much use if you can't actually make use of them. Finding sufficiently many different questions isn't all that easy.
From that point of view, I'd say the question you describe is also an attempt to introduce some of the thinking that is important in maths but often lacking in school maths lessons/exams. Rather than simply plugging numbers into a memorised algorithm, you need to actually conceptualise the question, to some degree at least, and apply the concepts in a different configuration.
Personally I'd happily take obscure over mind-numbingly-repetitive, but then I am a pure mathematician.
What do you mean by "mathematical training"? Or by "practically"?
Mathematics is used in different ways by different people; the form of training each should receive will vary and hopefully correspond to how they intend to use the results. However, creating patterns, recognizing patterns, and applying patterns are common themes across many of the forms that are practiced. If you want a general answer that may make sense, that answer would be that such problems should give practice in the creation, recognition, and application of patterns. If you want a general answer that is correct, that answer is that it depends on whether one is going to encounter items in arithmetic progression, or triples of tangents, or expressions in terms of signed sums of x,y, and z. Mathematicians (and other practitioners of mathematics) are making connections between the past and the present all the time, and it is often unclear how today's patterns will be used tomorrow.
Gerhard "Knowing The Future Is Boring" Paseman, 2015.04.14
