A surveyor wishes to measure the width of a trench. He places a transit telescope in a position on one side of the trench that is 18 feet higher than the second location. The surveyor reads the angle of depression from the transit telescope and finds that it is 31°, as shown in the diagram below. If the transit telescope is 4.5 feet above the ground at the first location during this reading, how wide is the trench?

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not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this