Question

The value of tan{sin⁻¹(3/5)+cos⁻¹(5/13)} is.......,Select Proper option from the given options.
(a) - 24/5
(b) - 22/15
(c) - 63/16
(d) - 47/12

Answer 1

please see the attachment

Answer 2

Answer:

(c) $-\frac{63}{16}$

Step-by-step explanation:

As we that that $tan\left ( A+B \right )=\frac{tanA+tanB}{1-tanA\times tanB}$

Now from the given question  $tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+ cos^{-1}\left ( \frac{5}{13} \right )\right )$

we have  $A= sin^{-1}\left ( \frac{3}{5} \right )$

              $\Rightarrow sinA= \left ( \frac{3}{5} \right )$

         ∴$\Rightarrow tanA= \left ( \frac{3}{4} \right )$

  and       $B= cos^{-1}\left ( \frac{5}{13} \right )$

             $\Rightarrow cosB= \left ( \frac{5}{13} \right )$

        ∴$\Rightarrow tanB= \left ( \frac{12}{5} \right )$

Firstly substitute the value of A and B in $tan\left ( A+B \right )=\frac{tanA+tanB}{1-tanA\times tanB}$

∴$\Rightarrow tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+cos^{-1}\left ( \frac{5}{13} \right ) \right )=\frac{tanA+tanB}{1-tanA\times tanB}$

Now,

Substitute the value of tanA and tanB in

$\Rightarrow tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+cos^{-1}\left ( \frac{5}{13} \right ) \right )=\frac{tanA+tanB}{1-tanA\times tanB}$

$\Rightarrow tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+cos^{-1}\left ( \frac{5}{13} \right ) \right )=\frac{\frac{3}{4}+\frac{12}{5}}{1-\frac{3}{4}\times \frac{12}{5}}$

$\Rightarrow tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+cos^{-1}\left ( \frac{5}{13} \right ) \right )=-\frac{63}{16}$

Hence ,the value of  $tan\left ( sin^{-1}\left ( \frac{3}{5} \right )+ cos^{-1}\left ( \frac{5}{13} \right )\right )=-\frac{63}{16}$