Question

Solve \[\dfrac{{\cot (90 - {\rm{\theta }})\sin (90 - {\rm{\theta }})}}{{\sin {\rm{\theta }}}} + \dfrac{{\cot 40}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]

Answer 1

Hint:
Here, we have to use the basic identities of the trigonometric functions to find out the value of the given equation. So we have to apply the properties of the function for the simplification of the equation and by solving the simplified equation we will get the value of the given equation.

Complete step by step solution:
So the given equation is \[\dfrac{{\cot (90 - {\rm{\theta }})\sin (90 - {\rm{\theta }})}}{{\sin {\rm{\theta }}}} + \dfrac{{\cot 40}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
So firstly we will take the first term i.e. \[\dfrac{{\cot (90 - {\rm{\theta }})\sin (90 - {\rm{\theta }})}}{{\sin {\rm{\theta }}}}\] and we have to simplifying this term by applying the basic identities of trigonometry. We know that
\[\cot (90 - {\rm{\theta }}) = \tan {\rm{\theta and }}\sin (90 - {\rm{\theta }}) = \cos {\rm{\theta }}\]
So by applying these identities we get the equation as
\[ \Rightarrow \dfrac{{\tan {\rm{\theta }}\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} + \dfrac{{\cot 40}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
Now simplify the first term i.e. \[\dfrac{{\tan {\rm{\theta }}\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}}\]in the above equation we get
\[ \Rightarrow 1 + \dfrac{{\cot 40}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
Now we have to take the second term\[\dfrac{{\cot 40}}{{\tan 50}}\] and in this term we can write the cot function in terms of tan function to simplify the equation. Then we get
\[ \Rightarrow 1 + \dfrac{{\cot (90 - 50)}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
And we know that\[\cot (90 - {\rm{\theta }}) = \tan {\rm{\theta }}\]therefore, we get
\[ \Rightarrow 1 + \dfrac{{\tan 50}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
\[ \Rightarrow 1 + 1 - \{ {(\cos 20)^2} + {(\cos 70)^2}\} \]
Now we will take the last term i.e. \[\{ {(\cos 20)^2} + {(\cos 70)^2}\} \]and simplify it. So we can write \[\cos 70 = \cos (90 - 20)\]and we know that \[\cos (90 - {\rm{\theta }}) = \sin {\rm{\theta }}\]therefore we get
\[ \Rightarrow 1 + 1 - \{ {(\cos 20)^2} + {(\cos (90 - 20))^2}\} \]
\[ \Rightarrow 1 + 1 - \{ {(\cos 20)^2} + {(\sin 20)^2}\} \]
And we know that\[{\rm{si}}{{\rm{n}}^2}{\rm{\theta }} + {\cos ^2}{\rm{\theta = 1}}\] so we get
\[ \Rightarrow 1 + 1 - \{ 1\} \]
\[ \Rightarrow 1\]
Therefore, \[\dfrac{{\cot (90 - {\rm{\theta }})\sin (90 - {\rm{\theta }})}}{{\sin {\rm{\theta }}}} + \dfrac{{\cot 40}}{{\tan 50}} - \{ {(\cos 20)^2} + {(\cos 70)^2}\} = 1\]

Hence, the value of the given equation is 1.

Note:
We should know the different properties of the trigonometric function and also in which quadrant which function is positive or negative as in the first quadrant all the functions i.e. sin, cos, tan, cot, sec, cosec is positive. In the second quadrant only sine and cosec function is positive and all the other functions are negative. In the third quadrant only tan and cot function is positive and in fourth quadrant only cos and sec function is positive. With the help of this concept this question can be easily solved.