11th Class Mathematics Trigonometric Identities Question Bank Critical Thinking

  • question_answer If \[a\,\cos 2\theta +b\,\sin 2\theta =c\]has a and b as its solution, then the value of \[\tan \alpha +\tan \beta \] is [Kurukshetra CEE 1998]

    A) \[\frac{c+a}{2b}\]

    B) \[\frac{2b}{c+a}\]

    C) \[\frac{c-a}{2b}\]

    D) \[\frac{b}{c+a}\]

    Correct Answer: B

    Solution :

    \[a\cos 2\theta +b\sin 2\theta =c\] Þ \[a\left( \frac{1-{{\tan }^{2}}\theta }{1+{{\tan }^{2}}\theta } \right)+b\frac{2\tan \theta }{1+{{\tan }^{2}}\theta }=c\] \[\Rightarrow \] \[a-a{{\tan }^{2}}\theta +2b\tan \theta =c+c{{\tan }^{2}}\theta \] \[\Rightarrow \]\[-(a+c){{\tan }^{2}}\theta +2b\,\tan \theta +(a-c)=0\] \[\therefore \tan \alpha +\tan \beta =-\frac{2b}{-(c+a)}=\frac{2b}{c+a}\] .

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