Step 1a

The given inverse is \$"cos"("tan"^-1(3/4))\$

Let, \$"tan"^-1(3/4) = θ\$

\$"tan"θ = 3/4\$

We know the formula that \$"sec"^2 θ - "tan"^2 θ = 1\$

⇒ sec θ = \$ \sqrt(1 + "tan"^2 θ)\$

Plug the value in the formula and then simplify
⇒ sec θ = \$\sqrt(1 + (3/4)^2)\$

⇒ sec θ = \$\sqrt(1 + 9)/(16)\$

⇒ sec θ = \$\sqrt(16 + 9) / 16\$

⇒ sec θ = \$\sqrt((25)/16)\$

⇒ \$"sec" θ = 5/4\$

Explanation: Use the inverse function, substitute the value into the formula, and simplify to find sec θ.

Step 1b

Therefore , \$"cos" θ = 4/5\$, \$"cos" θ = 1/("sec" θ)\$

⇒ \$θ = "cos"^-1 (4/5)\$

Now, \$"cos"("tan"^-1(3/4))\$ = \$"cos"("cos"^-1 (4/5)) = 4/5\$

Therefore, \$"cos"("tan"^-1 (3/4)) = 4/5\$.

Explanation: First, find cos θ. Then, determine the value of \$"cos"("tan"^-1(3/4))\$, which is \$4/5\$.