Mathematics

Mathematics

Problem

Simplify the following expression completely and then find its numerical value: a. (i) Simplify the numerator sin⁡(90∘−30∘) sec⁡30∘+cos⁡(90∘−30∘) csc⁡30∘\sin(90^{\circ}-30^{\circ})\,\sec30^{\circ}+\cos(90^{\circ}-30^{\circ})\,\csc30^{\circ}. b. (ii) Simplify the denominator tan⁡45∘+cot⁡45∘\tan45^{\circ}+\cot45^{\circ}. c. (iii) Hence, determine the value of the complete expression sin⁡(90∘−30∘) sec⁡30∘+cos⁡(90∘−30∘) csc⁡30∘tan⁡45∘+cot⁡45∘ .\frac{\sin(90^{\circ}-30^{\circ})\,\sec30^{\circ}+\cos(90^{\circ}-30^{\circ})\,\csc30^{\circ}}{\tan45^{\circ}+\cot45^{\circ}}\,.
Intuitive explanation

This problem asks you to take an algebraic expression and reduce it to its simplest form using the rules of exponents, radicals, or fractions—whichever apply—and then plug in any given values to get a single number. The key insight is to work systematically through one operation at a time: combine like terms, apply exponent rules (like am⋅an=am+na^m \cdot a^n = a^{m+n}), or rationalize denominators if needed, before substituting numbers. This order prevents messy arithmetic and makes errors easier to spot.

Intuitive explanation

This problem asks you to take an algebraic expression and reduce it to its simplest form using the rules of exponents, radicals, or fractions—whichever apply—and then plug in any given values to get a single number. The key insight is to work systematically through one operation at a time: combine like terms, apply exponent rules (like am⋅an=am+na^m \cdot a^n = a^{m+n}), or rationalize denominators if needed, before substituting numbers. This order prevents messy arithmetic and makes errors easier to spot.

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