Given that `tantheta=1/(sqrt(3,))` the value of `(cos e c^2theta-sec^2theta)/(cos e c^2theta+sec^2theta)` is `-1` (b) `1` (c) `1/2` (d) `-1/2`

Explanation

`tan theta=frac{1}{sqrt{3}} Rightarrow tan theta=tan 30^{circ} Rightarrow theta=30^{circ} `


`frac{{cosec}^{2} theta-sec ^{2} theta}{{cosec}^{2} theta+sec ^{2} theta}=frac{{cosec}^{2} 30^{circ}-sec ^{2} 30^{circ}}{{cosec}^{2} 30^{circ}+sec ^{2} 30^{circ}} `


`=frac{2^{2}-(2 / sqrt{3})^{2}}{2^{2}+(2 / sqrt{3})^{2}} `


`=frac{4-4 / 3}{4+4 / 3}=frac{12-4}{12+4}=frac{8}{16}=frac{1}{2}`