Question

Evaluate the following
\[\left( {{\operatorname{cosec}}^{2}}{{45}^{o}}{{\sec }^{2}}{{30}^{o}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\]

Answer 1

Hint:First of all, consider the expression given in the question. Now make the table for trigonometric ratios of general angles. Now, from that find the values of \[\sin {{30}^{o}},\operatorname{cosec}{{45}^{o}},\sec {{60}^{o}},\sec {{30}^{o}}\] and \[\cot {{45}^{o}}\] and substitute these in the given expression to get the required answer.

Complete step-by-step answer:
In this question, we have to find the value of the expression
\[\left( {{\operatorname{cosec}}^{2}}{{45}^{o}}{{\sec }^{2}}{{30}^{o}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\]
Let us consider the expression given in the question.
\[E=\left( {{\operatorname{cosec}}^{2}}{{45}^{o}}{{\sec }^{2}}{{30}^{o}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)....\left( i \right)\]
Now, we have to find the values of \[\sin {{30}^{o}},\operatorname{cosec}{{45}^{o}},\sec {{60}^{o}},\sec {{30}^{o}}\] and \[\cot {{45}^{o}}\].
Let us make the table for trigonometric ratios of general angles like \[{{0}^{o}},{{30}^{o}},{{45}^{o}},{{60}^{o}},{{90}^{o}}\] and find the required values.

From the above table, we get, \[\operatorname{cosec}{{45}^{o}}=\sqrt{2}\]. By substituting this in equation (i), we get,
\[E=\left( {{\left( \sqrt{2} \right)}^{2}}.{{\sec }^{2}}{{30}^{o}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\]
Also from the above table, we get \[\sec {{30}^{o}}=\dfrac{2}{\sqrt{3}}\]. By substituting this in the above equation, we get, \[E=\left( {{\left( \sqrt{2} \right)}^{2}}.{{\left( \dfrac{2}{\sqrt{3}} \right)}^{2}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\]
From the table, we also get, \[\sin {{30}^{o}}=\dfrac{1}{2}\]. By substituting this in the above equation, we get,
\[E=\left( {{\left( \sqrt{2} \right)}^{2}}.{{\left( \dfrac{2}{\sqrt{3}} \right)}^{2}} \right)\left( {{\left( \dfrac{1}{2} \right)}^{2}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\]
From the table, we also get, \[\cot {{45}^{o}}=1\]. By substituting this in the above equation, we get,
\[E=\left( {{\left( \sqrt{2} \right)}^{2}}.{{\left( \dfrac{2}{\sqrt{3}} \right)}^{2}} \right)\left( {{\left( \dfrac{1}{2} \right)}^{2}}+4{{\left( 1 \right)}^{2}}-{{\sec }^{2}}{{60}^{o}} \right)\]
From the table, we also get, \[\sec {{60}^{o}}=2\]. By substituting this in the above equation, we get,
\[E=\left( {{\left( \sqrt{2} \right)}^{2}}.{{\left( \dfrac{2}{\sqrt{3}} \right)}^{2}} \right)\left( {{\left( \dfrac{1}{2} \right)}^{2}}+4{{\left( 1 \right)}^{2}}-{{\left( 2 \right)}^{2}} \right)\]
By simplifying the above equation, we get,
\[E=\left[ 2\left( \dfrac{4}{3} \right) \right]\left[ \dfrac{1}{4}+4-4 \right]\]
\[E=\left( \dfrac{8}{3} \right)\left( \dfrac{1}{4} \right)\]
\[E=\dfrac{2}{3}\]
Hence, we get the value of the expression \[\left( {{\operatorname{cosec}}^{2}}{{45}^{o}}{{\sec }^{2}}{{30}^{o}} \right)\left( {{\sin }^{2}}{{30}^{o}}+4{{\cot }^{2}}{{45}^{o}}-{{\sec }^{2}}{{60}^{o}} \right)\] as \[\dfrac{2}{3}\].

Note: In these types of questions, students just need to remember the values of \[\sin \theta \] and \[\cos \theta \] at various angles like \[{{0}^{o}},{{30}^{o}},{{60}^{o}},{{45}^{o}},\] etc. and they can find all other trigonometric ratios using them. For example, they can find \[\operatorname{cosec}{{45}^{o}}\] by using \[\dfrac{1}{\sin {{45}^{o}}},\sec {{30}^{o}}\] by using \[\dfrac{1}{\cos {{30}^{o}}},\cot {{45}^{o}}\] by using \[\dfrac{\cos {{45}^{o}}}{\sin {{45}^{o}}}\] and \[\sec {{60}^{o}}\] by using \[\dfrac{1}{\cos {{60}^{o}}}\].