Question

Find the exact value of the expression whenever it is defined?

tan(arctan 6/5 + arccos 7/25)

Answer 1

`-162/109` or `-78/179`

Explanation:

We use the formula

`tan(A+B) = (tanA+tanB)/(1-tanAtanB)`

In this problem

`A = tan^-1(6/5) implies tanA = 6/5`

and

`B = cos^-1(7/25) implies sec B = 25/7 implies`

`tan^2 B = sec^2B-1 = (25/7)^2-1=576/49 implies`

`tan^2B = (24/7)^2`

So, `tanB = +24/7` or `tan B = -24/7`

If `tan B = +24/7`, we have

`tan(A+B) = (6/5+24/7)/(1-6/5 times 24/7) = (6 times 7+24 times 5)/(5 times 7-6times 24) = -162/109`

On the other hand, if `tan B = -24/7`, we have

`tan(A+B) = (6/5-24/7)/(1-6/5 times (-24/7)) = (6 times 7-24 times 5)/(5 times 7+6times 24) = -78/179`