QUESTION

Find the value of following trigonometric function:$\dfrac{{\sin {{35}^0}}}{{\cos {{55}^0}}} + \dfrac{{\tan {{37}^0}}}{{\cot {{53}^0}}}$

Hint: Try to break the angle as a sum of other angles with multiple of ${90^0},{180^0},{270^0} \& {360^0}$. In order to solve this question we will use trigonometric identities such as
$\cos (90 - \theta ) = \sin \theta {\text{ and }}\cot (90 - \theta ) = \tan \theta$

Given term is: $\dfrac{{\sin {{35}^0}}}{{\cos {{55}^0}}} + \dfrac{{\tan {{37}^0}}}{{\cot {{53}^0}}}$
In order to make the term simpler, we will use the trigonometric identities to convert $cos {\theta}$ into $sin{\theta}$ and $cot{\theta}$ into $tan{\theta}$
$\cos (90 - \theta ) = \sin \theta {\text{ and }}\cot (90 - \theta ) = \tan \theta$
$\Rightarrow \dfrac{{\sin {{35}^0}}}{{\cos {{55}^0}}} + \dfrac{{\tan {{37}^0}}}{{\cot {{53}^0}}} \\ \Rightarrow \dfrac{{\sin {{35}^0}}}{{\cos \left( {{{90}^0} - {{35}^0}} \right)}} + \dfrac{{\tan {{37}^0}}}{{\cot \left( {{{90}^0} - {{37}^0}} \right)}} \\ \Rightarrow \dfrac{{\sin {{35}^0}}}{{\sin {{35}^0}}} + \dfrac{{\tan {{37}^0}}}{{\tan {{37}^0}}} \\$
$\Rightarrow \dfrac{{\sin {{35}^0}}}{{\sin {{35}^0}}} + \dfrac{{\tan {{37}^0}}}{{\tan {{37}^0}}} \\ = 1 + 1 \\ = 2 \\$
Hence, the value of given term $\dfrac{{\sin {{35}^0}}}{{\cos {{55}^0}}} + \dfrac{{\tan {{37}^0}}}{{\cot {{53}^0}}}$ is 2.