$$\begin{gathered} {\mathrm{cosec}}^2 57° - {\tan }^2 33° + \cos 44° \mathrm{cosec} 46° - \sqrt{2} \cos 45° - {\tan }^2 60° \\ = {\mathrm{cosec}}^2 ( 90° - 33° )° - {\tan }^2 33° + \cos 44° \mathrm{cosec} ( 90° -44 )° - \sqrt{2} \cos 45° - {\tan }^2 60° \\ = {\sec }^{2 }33° - {\tan }^2 33° + \cos 44° \sec 44° - \sqrt{2} \cos 45° - {\tan }^2 60° \\ = 1 + 1 - \sqrt{2} \cos 45° - {\tan }^2 60° \\ = 1 + 1 - \sqrt{2} \left( \frac{1}{\sqrt{2}} \right) - {\left( \sqrt{3} \right)}^2 \\ = 2 - 1 - 3 \\ = - 2 \end{gathered}$$

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}. = \frac{\cos \mathrm{A}}{1 + \sin \mathrm{A}} + \tan \mathrm{A} \\ = \frac{\cos \mathrm{A} ( 1 - \sin \mathrm{A} )}{( 1 + \sin \mathrm{A} ) ( 1 - \sin \mathrm{A} )} + \frac{\sin \mathrm{A}}{\cos \mathrm{A}} \\ = \frac{\cos \mathrm{A} - \sin \mathrm{A} \cos \mathrm{A} )}{ 1^2 - {\sin }^2 \mathrm{A} } + \frac{\sin \mathrm{A}}{\cos \mathrm{A}} \\ = \frac{\cos \mathrm{A} - \sin \mathrm{A} \cos \mathrm{A} )}{ 1 - {\sin }^2 \mathrm{A} } + \frac{\sin \mathrm{A}}{\cos \mathrm{A}} \\ = \frac{\cos \mathrm{A} - \sin \mathrm{A} \cos \mathrm{A} )}{ {\cos }^2 \mathrm{A} } + \frac{\sin \mathrm{A}}{\cos \mathrm{A}} \\ = \frac{1}{\cos \mathrm{A}} - \frac{\sin \mathrm{A}}{\cos \mathrm{A}} + \frac{\sin \mathrm{A}}{\cos \mathrm{A}} \\ = \frac{1}{\cos \mathrm{A}} \end{gathered}$$

= sec A

= R.H.S.