Question

Fill in the blank: ${{\sin }^{-1}}x+{{\cos }^{-1}}x=\_\_\_\_\_\_\_$ .

Answer 1

Hint: To find the value of ${{\sin }^{-1}}x+{{\cos }^{-1}}x$ , we will first create an equation by equating ${{\sin }^{-1}}x$ to $\theta $ . Then from this equation, we will find x which will be equal to $\sin \theta $ . We will then use the formula $\cos \left( \dfrac{\pi }{2}-\theta \right)=\sin \theta $ and substitute this in the previous equation. After a few rearrangements and substitutions, we will get the required solution.

Complete step-by-step solution:
We have to find the value of ${{\sin }^{-1}}x+{{\cos }^{-1}}x$ . Let us consider
${{\sin }^{-1}}x=\theta ...\left( i \right)$
From the above equation, we can write,
$x=\sin \theta $
We know that $\cos \left( \dfrac{\pi }{2}-\theta \right)=\sin \theta $ . Therefore, we can write the above equation as
$\Rightarrow x=\cos \left( \dfrac{\pi }{2}-\theta \right)$
We can take the cos to the LHS, so that the above equation can be written as
$\Rightarrow {{\cos }^{-1}}x=\left( \dfrac{\pi }{2}-\theta \right)$
Let us substitute equation (i) in the above equation. Hence, we will get
$\Rightarrow {{\cos }^{-1}}x=\left( \dfrac{\pi }{2}-{{\sin }^{-1}}x \right)$
Now, we have to move ${{\sin }^{-1}}x$ to the LHS. Then, we can write the above equation as
$\Rightarrow {{\cos }^{-1}}x+{{\sin }^{-1}}x=\dfrac{\pi }{2}$
Therefore, ${{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}$ .

Note: Students must be thorough with all the rules , formulas and properties of trigonometric functions. They must also be thorough with the arc functions or inverse trigonometric functions. The inverse trigonometric functions perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. We usually call inverse trigonometric functions as arc functions, for example, for ${{\sin }^{-1}}x$ we can write arcsin, for ${{\cos }^{-1}}x$ , we can write arcos and so on. Students must know to solve algebraic equations and associated rules and properties.
The above proved property is applicable for x in the range $-1\le x\le 1$ or for $x\in \left[ -1,1 \right]$ .