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Mathematics

Evaluate the following :

sin 30° + tan 45° - cosec 60°sec 30° + cos 60° + cot 45°\dfrac{\text{sin 30° + tan 45° - cosec 60°}}{\text{sec 30° + cos 60° + cot 45°}}

Trigonometric Identities

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Answer

Substituting values, we get :

⇒sin 30° + tan 45° - cosec 60°sec 30° + cos 60° + cot 45°=12+1−2323+12+1=32−2332+23=32−2332+23×32−2332−23=(32−23)2(32)2−(23)2=(32)2+(23)2−2×32×23(32)2−(23)2=94+43−2394−43=27+16−2431227−1612=43−24311.\Rightarrow \dfrac{\text{sin 30° + tan 45° - cosec 60°}}{\text{sec 30° + cos 60° + cot 45°}} = \dfrac{\dfrac{1}{2} + 1 - \dfrac{2}{\sqrt{3}}}{\dfrac{2}{\sqrt{3}} + \dfrac{1}{2} + 1} \\[1em] = \dfrac{\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}}{\dfrac{3}{2} + \dfrac{2}{\sqrt{3}}} \\[1em] = \dfrac{\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}}{\dfrac{3}{2} + \dfrac{2}{\sqrt{3}}} \times \dfrac{\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}}{\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}} \\[1em] = \dfrac{\Big(\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}\Big)^2}{\Big(\dfrac{3}{2}\Big)^2 - \Big(\dfrac{2}{\sqrt{3}}\Big)^2} \\[1em] = \dfrac{\Big(\dfrac{3}{2}\Big)^2 + \Big(\dfrac{2}{\sqrt{3}}\Big)^2 - 2 \times \dfrac{3}{2} \times \dfrac{2}{\sqrt{3}}}{\Big(\dfrac{3}{2}\Big)^2 - \Big(\dfrac{2}{\sqrt{3}}\Big)^2} \\[1em] = \dfrac{\dfrac{9}{4} + \dfrac{4}{3} - 2\sqrt{3}}{\dfrac{9}{4} - \dfrac{4}{3}} \\[1em] = \dfrac{\dfrac{27 + 16 - 24\sqrt{3}}{12}}{\dfrac{27 - 16}{12}} \\[1em] = \dfrac{43 - 24\sqrt{3}}{11}.

Hence, sin 30° + tan 45° - cosec 60°sec 30° + cos 60° + cot 45°=43−24311.\dfrac{\text{sin 30° + tan 45° - cosec 60°}}{\text{sec 30° + cos 60° + cot 45°}} = \dfrac{43 - 24\sqrt{3}}{11}.

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