Let [ ( + ) = 4 5] and [ ,,,( - ) = 5 13,] where [0 ,, 4.] Then [tan,2 ] =

Question

Let $$\cos \left( {\alpha  + \beta } \right) = {4 \over 5}$$ and $$\sin \,\,\,\left( {\alpha  - \beta } \right) = {5 \over {13}},$$ where $$0 \le \alpha ,\,\beta  \le {\pi  \over 4}.$$

Then $$tan\,2\alpha $$ =

Accepted Answer

$$\cos \left( {\alpha  + \beta } \right) = {4 \over 5} \Rightarrow \tan \left( {\alpha  + \beta } \right) = {3 \over 4}$$

$$\sin \left( {\alpha  - \beta } \right) = {5 \over {13}} \Rightarrow \tan \left( {\alpha  - \beta } \right) = {5 \over {12}}$$

$$\tan 2\alpha  = \tan \left[ {\left( {\alpha  + \beta } \right) + \left( {\alpha  - \beta } \right)} \right]$$

$$ = {{{3 \over 4} + {5 \over {12}}} \over {1 - {3 \over 4}.{5 \over {12}}}} = {{56} \over {33}}$$

Solutions · 1