Question

Find the value of ${\cos ^2}\theta \left( {1 + {{\tan }^2}\theta } \right) + {\sin ^2}\theta \left( {1 + {{\cot }^2}\theta } \right)$
$\left( A \right)$ 1
$\left( B \right)$ 2
$\left( C \right)$ 3
$\left( D \right)$ 4

Answer 1

Hint – In this question use the concept that tan is the ratio of sine to cosine and cot is the ratio of cosine to sin (i.e. tan = sin/cos, cot = cos/sin) and later on in the solution use the basic trigonometric equation \[\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) = 1\], so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given trigonometric equation is
${\cos ^2}\theta \left( {1 + {{\tan }^2}\theta } \right) + {\sin ^2}\theta \left( {1 + {{\cot }^2}\theta } \right)$
Now as we know that tan is the ratio of sine to cosine and cot is the ratio of cosine to sin (i.e. tan = sin/cos, cot = cos/sin) so use this property in the above equation we have,
$ \Rightarrow {\cos ^2}\theta \left( {1 + \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}} \right) + {\sin ^2}\theta \left( {1 + \dfrac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)$
Now take the L.C.M of the denominator of the above equation we have,
$ \Rightarrow {\cos ^2}\theta \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }}} \right) + {\sin ^2}\theta \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)$
Now cancel out the common factors from denominator and the numerator we have,
$ \Rightarrow \left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) + \left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)$
Now again simplify it we have,
$ \Rightarrow 2\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)$
Now as we know that \[\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)\] is a trigonometric identity whose value is always one so use this property in the above equation we have,
$ \Rightarrow 2\left( 1 \right) = 2$
So 2 is the required answer of the given trigonometric equation.
So this is the required answer.
Hence option (C) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that always recall the basic trigonometric properties which is very helpful to get on the right track to get the answer of the problem, and the properties which is used in this problem is all stated above then simplify the given trigonometric equation using this basic trigonometric properties as above we will get the required solution of the given trigonometric equation.