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If  \frac{{\sin \left( {\theta + \alpha } \right)}}{{\cos \left( {\theta - \alpha } \right)}} = \frac{{1 - m}}{{1 + m}}, Find the value of  \tan\left( {\frac{\pi }{4} - \theta } \right).\tan\left( {\frac{\pi }{4} - \alpha } \right) 

  • Option 1)

    -m

  • Option 2)

    m

  • Option 3)

    -1/m

  • Option 4)

    1/m

 

Answers (1)

best_answer

As we learnt

 

Subtraction Formulae -

 

\tan \left ( A-B \right )= \frac{\tan A-\tan B}{1+\tan A\tan B}

- wherein

A and B are two angles.

 

 

 

Given, \frac{{\sin \left( {\theta + \alpha } \right)}}{{\cos \left( {\theta - \alpha } \right)}}$ = \frac{{1 - m}}{{1 + m}}$

            or, \frac{{\sin \left( {\theta + \alpha } \right) + \cos \left( {\theta - \alpha } \right)}}{{\sin \left( {\theta + \alpha } \right) - \cos \left( {\theta - \alpha } \right)}}$ = \frac{{1 - m + 1 + m}}{{1 - m - 1 - m}}$  [by componendo and dividendo].

            or, \frac{{\sin \left( {\theta + \alpha } \right) + \sin \left[ {\frac{\pi }{2} - \left( {\theta - \alpha } \right)} \right]}}{{\sin \left( {\theta + \alpha } \right) - \sin \left[ {\frac{\pi }{2} - \left( {\theta - \alpha } \right)} \right]}}$ = \frac{2}{{ - 2m}} = - \frac{1}{m}$

            or, \frac{{2\sin \frac{{\theta + \alpha + \frac{\pi }{2} - \theta + \alpha }}{2}}}{{2\cos \frac{{\theta + \alpha + \frac{\pi }{2} - \theta + \alpha }}{2}}}\frac{{\cos \frac{{\theta + \alpha - \frac{\pi }{2} + \theta - \alpha }}{2}}}{{\sin \frac{{\theta + \alpha - \frac{\pi }{2} + \theta - \alpha }}{2}}} = - \frac{1}{m}$

            or, \frac{{\sin \left( {\frac{\pi }{4} + \alpha } \right).\cos \left( {\theta - \frac{\pi }{4}} \right)}}{{\cos \left( {\frac{\pi }{4} + \alpha } \right).\sin \left( {\theta - \frac{\pi }{4}} \right)}} = - \frac{1}{m}$     or, tan\left( {\frac{\pi }{4} + \alpha } \right)$.cot\left( {\theta - \frac{\pi }{4}} \right)$ = -\frac{1}{m}$

            or, -tan\left( {\frac{\pi }{4} + \alpha } \right)$cot\left( {\frac{\pi }{4} - \theta } \right)$ = -\frac{1}{m}$                          [Qcot(-q) = -cotq]

            or,  m   = cot\left( {\frac{\pi }{4} + \alpha } \right)\tan \left( {\frac{\pi }{4} - \theta } \right)$

                  m   = tan\left[ {\frac{\pi }{2} - \left( {\frac{\pi }{4} + \alpha } \right)} \right]\tan \left( {\frac{\pi }{4} - \theta } \right)$         = tan\left( {\frac{\pi }{4} + \alpha } \right)\tan \left( {\frac{\pi }{4} - \theta } \right)$


Option 1)

-m

Option 2)

m

Option 3)

-1/m

Option 4)

1/m

Posted by

Himanshu

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