Question

How do you find the exact value without a calculator $1 - {\tan ^2}{10^ \circ } + {\csc ^2}{80^ \circ }$?

Answer 1

Hint: First of all, convert the tangent of 10 in cotangent using the fact that $\tan \left( {{{90}^ \circ } - x} \right) = \cot x$. Then, just use the fact that ${\csc ^2}\theta - {\cot ^2}\theta = 1$, thus the answer will be equal to 2.

Complete step by step solution:
We are given that we are required to find the exact value without a calculator $1 - {\tan ^2}{10^ \circ } + {\csc ^2}{80^ \circ }$.
Since, we know that we have an identity given by the following expression with us:-
$ \Rightarrow \tan \left( {{{90}^ \circ } - x} \right) = \cot x$
Replacing x by ${80^ \circ }$ in the above mentioned expression, we will then obtain the following expression with us:-
\[ \Rightarrow \tan \left( {{{90}^ \circ } - {{80}^ \circ }} \right) = \cot {80^ \circ }\]
Simplifying the quantities inside the bracket on the left hand side of the above mentioned expression, we will then obtain the following expression with us:-
\[ \Rightarrow \tan {10^ \circ } = \cot {80^ \circ }\]
Putting this in the expression given to us, we will then obtain the following expression with us:-
$ \Rightarrow 1 - {\tan ^2}{10^ \circ } + {\csc ^2}{80^ \circ } = 1 - {\cot ^2}{80^ \circ } + {\csc ^2}{80^ \circ }$ ……………….(1)
Now, since we know that we have an identity given by the following expression with us:-
$ \Rightarrow {\csc ^2}\theta - {\cot ^2}\theta = 1$
Replacing $\theta $ by ${80^ \circ }$ in the above mentioned expression, we will then obtain the following expression with us:-
$ \Rightarrow {\csc ^2}{80^ \circ } - {\cot ^2}{80^ \circ } = 1$
Putting this in equation number 1, we will then obtain the following expression with us:-
$ \Rightarrow 1 - {\tan ^2}{10^ \circ } + {\csc ^2}{80^ \circ } = 1 + 1$
Simplifying the right hand side of the above expression, we will then obtain the following expression with us:-
$ \Rightarrow 1 - {\tan ^2}{10^ \circ } + {\csc ^2}{80^ \circ } = 2$
Hence, the required answer is 2.

Note:
The students must note that we have used the following facts and formulas in the above solution:-
$\tan \left( {{{90}^ \circ } - x} \right) = \cot x$
${\csc ^2}\theta - {\cot ^2}\theta = 1$
The students must also note that we can also derive the formula in second point as follows:-
Since, we know that we have an identity given by the following expression:-
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1$
Dividing both the sides of the above equation by ${\sin ^2}\theta $, we will then obtain the following expression with us:-
$ \Rightarrow 1 + {\cot ^2}\theta = {\sec ^2}\theta $
Taking ${\cot ^2}\theta $ from addition in the left hand side to subtraction in the right hand side, we will then obtain the following expression with us:-
$ \Rightarrow {\csc ^2}\theta - {\cot ^2}\theta = 1$