| The value of \[\frac{1-{{\sin }^{2}}(\theta +16{}^\circ )}{1+{{\sin }^{2}}(\theta +31{}^\circ )}\] \[\times \frac{{{\cos }^{2}}(\theta +46{}^\circ )+{{\cos }^{2}}(\theta +16{}^\circ )}{\text{cose}{{\text{c}}^{2}}(\theta +76{}^\circ )-{{\cot }^{2}}(\theta +76{}^\circ )}\] \[\div \,\{\sin (\theta +46{}^\circ )\tan (\theta +16{}^\circ )\}\]for \[\theta =14{}^\circ \]is [SSC (CGL) 2014] |
A) \[-1\]
B) \[0\]
C) \[\frac{1}{2}\]
D) \[1\]
Correct Answer: D
Solution :
| \[\frac{1-{{\sin }^{2}}(\theta +16{}^\circ )}{1+{{\sin }^{2}}(\theta +31{}^\circ )}\times \frac{{{\cos }^{2}}(\theta +46{}^\circ )+{{\cos }^{2}}(\theta +16{}^\circ )}{\begin{align} & \text{cose}{{\text{c}}^{2}}(\theta +76{}^\circ )-{{\cot }^{2}}(\theta +76{}^\circ ) \\ & \div \,\,\{\sin (\theta +46{}^\circ )\tan (\theta +16{}^\circ )\} \\ \end{align}}\] |
| \[=\frac{1-{{\sin }^{2}}(14{}^\circ +16{}^\circ )}{1+{{\sin }^{2}}(14{}^\circ +31{}^\circ )}\times \frac{{{\cos }^{2}}(14{}^\circ +46{}^\circ )+{{\cos }^{2}}(14{}^\circ +16{}^\circ )}{\text{cose}{{\text{c}}^{2}}(14{}^\circ +76{}^\circ )-{{\cot }^{2}}(14{}^\circ +76{}^\circ )}\]\[\times \frac{1}{\sin (14{}^\circ +46{}^\circ )\tan (14{}^\circ +76{}^\circ )}\] |
| \[[putting\theta =14{}^\circ ]\] |
| \[=\frac{1-{{\sin }^{2}}30{}^\circ }{1+{{\sin }^{2}}45{}^\circ }\times \frac{{{\cos }^{2}}60{}^\circ +{{\cos }^{2}}30{}^\circ }{\text{cose}{{\text{c}}^{2}}90{}^\circ -{{\cot }^{2}}90{}^\circ }\times \frac{1}{\sin 60{}^\circ \tan 30{}^\circ }\] |
| \[=\frac{1-{{\left( \frac{1}{2} \right)}^{2}}}{1+{{\left( \frac{1}{\sqrt{2}} \right)}^{2}}}\times \frac{{{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{\sqrt{3}}{2} \right)}^{2}}}{{{(1)}^{2}}-{{\left( \frac{1}{\infty } \right)}^{2}}}\times \frac{1}{\left( \frac{\sqrt{3}}{2} \right)\cdot \left( \frac{1}{\sqrt{3}} \right)}\] |
| \[=\frac{1-\frac{1}{4}}{1+\frac{1}{2}}\times \frac{\frac{1}{4}+\frac{3}{4}}{1-0}\times \frac{1}{\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{3}}}\] |
| \[=\frac{3/4}{3/2}\times \frac{1}{1}\times 2=\frac{1}{2}\times 2=1\] |
You need to login to perform this action.
You will be redirected in
3 sec
