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SSC Sample Paper Mock Test-2 SSC CGL Tear-II Paper-1

  • question_answer
    If A \[=\frac{\pi }{6}\]and B \[=\frac{\pi }{3},\] then which of the following is/are correct? I. \[\sin A+\sin B=\cos A+\cos B\] II. \[\tan A+\tan B=\cot A+\cot B\] Select the correct answer using the code given below.

    A)  Only I             

    B)  Only II

    C)  Both I and II    

    D)  Neither I nor II

    Correct Answer: C

    Solution :

    Given, A \[\frac{\pi }{6}\]and B\[\frac{\pi }{3}\]
    I. LHS,                   
    \[\sin A+\sin B=\sin \frac{\pi }{6}+\sin \frac{\pi }{3}=\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2}\]
    RHS, \[\cos A+\cos B=\cos \frac{\pi }{6}+\cos \frac{\pi }{3}=\frac{\sqrt{3}}{2}+\frac{1}{2}=\frac{\sqrt{3}+1}{2}\]\[\Rightarrow \]\[\sin A+\sin B=\cos A+\cos B\]
    II. LHS, \[\tan A+\tan B=\tan \frac{\pi }{6}+\tan \frac{\pi }{3}\]
    \[=\frac{1}{\sqrt{3}}+\sqrt{3}=\frac{4}{\sqrt{3}}\]
    RHS, \[\cot A+\cot B=\cot \frac{\pi }{6}+\cot \frac{\pi }{3}=\sqrt{3}+\frac{1}{\sqrt{3}}=\frac{4}{\sqrt{3}}\]
    \[\Rightarrow \]\[\tan \,\,A+tan\,\,B=cot\,\,A+cot\,\,B\]
    Hence, both statements are true.
    Shortcut method
    \[A+B=\frac{\pi }{6}+\frac{\pi }{3}=\frac{\pi }{2}\]
    I. \[\sin A+\sin B=\sin \left( \frac{\pi }{2}-B \right)+\sin \left( \frac{\pi }{2}-A \right)\]
    \[=\cos B+\cos A=\cos A+\cos B\]
    II. \[\tan A+\tan B=\tan \left( \frac{\pi }{2}-B \right)+\tan \left( \frac{\pi }{2}-A \right)\]
    \[=\cot B+\cot A=\cot A+\cot B\]
    \[\therefore \]Both statements are true.
               


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