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Question: Evaluate the following: \(2{\tan ^2}45^\circ + {\cos ^2}30^\circ - {\sin ^2}60^\circ \)...

Evaluate the following: 2tan⁡245∘+cos⁡230∘−sin⁡260∘2{\tan ^2}45^\circ + {\cos ^2}30^\circ - {\sin ^2}60^\circ

Explanation

Solution

Hint : Take the given expression and place the equivalent values using the trigonometric table and simplify the expression by using the squares of the number and least common multiple and simplify the resultant required value.

Complete step-by-step answer :
Take the given expression: 2tan⁡245∘+cos⁡230∘−sin⁡260∘2{\tan ^2}45^\circ + {\cos ^2}30^\circ - {\sin ^2}60^\circ
We have
tan⁡245∘=1{\tan ^2}45^\circ = 1
cos⁡230∘=(32)2{\cos ^2}30^\circ = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2}
sin⁡260∘=(32)2{\sin ^2}60^\circ = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2}
Place the values in the above expression by using the trigonometric table for the different functions and its angles.
=2(1)+(32)2−(32)2= 2(1) + {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} - {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2}
Since cos⁡230∘{\cos ^2}30^\circ and sin⁡260∘{\sin ^2}60^\circ reads the same value they cancel out.
∴2(1)+0=2\therefore 2(1) + 0 = 2
So, the correct answer is “2”.

Note : Remember the trigonometric table for different trigonometric functions and its angles and place the value carefully since it is the basic and important step for the solution. Also remember that when a term is multiplied with itself it gives the square of the number. Be careful about the sign convention while simplifying.