Given:
The trigonometric expressions are given as,
[tex]\begin{gathered} a)\text{ }(\cos 60\degree)(\sin 270\degree)+\tan 225\degree \\ b)-\tan 240\degree+(cos45\degree)(\sec 135\degree) \end{gathered}[/tex]Explanation:
a)
The given expression can be rewritten as,
[tex]=(\cos 60\degree)(\sin (360\degree-90\degree))+\tan (270\degree-45\degree)\text{ . . . . .(1)}[/tex]Since, from the trigonometric ratios,
[tex]\begin{gathered} \sin (360\degree-90\degree)=-\sin 90\degree \\ \tan (270\degree-45\degree)=\cot 45\degree \end{gathered}[/tex]On plugging the obtained ratios in equation (1),
[tex]=(\cos 60\degree)(-\sin 90\degree)+\cot 45[/tex]Substitute the trigonometric values in the above equation.
[tex]\begin{gathered} =\frac{1}{2}(-1)+1 \\ =-\frac{1}{2}+1 \\ =\frac{1}{2} \end{gathered}[/tex]Hence, the exact value of the expression is 1/2.
b)
The given expression can be rewritten as,
[tex]=-\tan (270\degree-30\degree)+(\cos 45\degree)(\sec (90\degree+45\degree))\text{ . . . ..(2)}[/tex]Since, from the trigonometric ratios,
[tex]\begin{gathered} \tan (270\degree-30\degree)=\cot 30\degree \\ \sec (90\degree+45\degree)=-\csc 45\degree \end{gathered}[/tex]On plugging the obtained ratios in equation (2),
[tex]=-\cot 30\degree+(\cos 45\degree)(-\csc 45)[/tex]Substitute the trigonometric values in the above equation.
[tex]\begin{gathered} =-\sqrt[]{3}+\frac{1}{\sqrt[]{2}}(-\sqrt[]{2}) \\ =-\sqrt[]{3}-1 \end{gathered}[/tex]Hence, the exact value of the expression is -√3-1.
