Question

How do you simplify $\tan ({\sin ^{ - 1}}(x))?$

Answer 1

Hint:
Whenever such a type of question you have to solve, always assume $()$ value to be x or θ, because it is always easy to solve. After that use Pythagorean Theorem.

Complete step by step solution:
As, mentioned in hint for solving such type of question, we will first assume ${\sin ^{ - 1}}x = \theta $
So, $\sin \theta = x$
Now we can rewrite the above equation as $\sin \theta = \dfrac{x}{1}$.
For $0 < x < 1$, if we draw right triangle having Hypotenuse one and opposite x,
Then, $\sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$
Now applying Pythagorean Theorem for the right triangle having Hypotenuse one and opposite x, we get the other side of the triangle.
Which is $\sqrt {1 - {x^2}} $.
Now, we already know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ and $\cos \theta = \sqrt {1 - {{\sin }^2}\theta } $
Now substituting value of $\cos \theta $ in $\tan \theta $ equation we get,
$\tan \theta = \dfrac{{\sin \theta }}{{\sqrt {1 - {{\sin }^2}\theta } }}$
Now, we will substitute value of θ in $\tan \theta $ equation,
So after substitute value of θ in $\tan \theta $ equation, we get,

$\tan ({\sin ^{ - 1}}x) = \dfrac{x}{{\sqrt {1 - {x^2}} }}$

Note:
In such types of questions newer confuse or scare. If you just understand one question perfectly and after that practice one or two questions, you are able to solve any difficult question of such type. Just remember some basic trigonometry formulas and some basic rules. Always try to solve in a proper step by step method because if you have done any mistake there it is very difficult to find.