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Question: How do I simplify \(\sin \left( {\left( {\tan ^{- 1}} \right)x} \right)\)?...

How do I simplify sin⁡((tan⁡−1)x)\sin \left( {\left( {\tan ^{- 1}} \right)x} \right)?

Explanation

Solution

In order to determine the simplification of the above question, let the tan⁡−1x=θ{\tan ^{ - 1}}x = \theta , and recall that sin⁡θ\sin \theta can be written as tan⁡θsec⁡θ\dfrac{{\tan \theta }}{{\sec \theta }}. Now replacing the sec⁡θ\sec \theta in the denominator using the identity of trigonometry which state that sec⁡2θ=tan⁡2θ+1{\sec ^2}\theta = {\tan ^2}\theta + 1 and after putting this put x=tan⁡θx = \tan \theta as we have assumed earlier.

Formula used:
sin⁡θ=tan⁡θsec⁡θ\sin \theta = \dfrac{{\tan \theta }}{{\sec \theta }}
sec⁡2θ=tan⁡2θ+1{\sec ^2}\theta = {\tan ^2}\theta + 1

Complete step by step answer:
We are given a trigonometric expression sin⁡(tan⁡−1(x))\sin ({\tan ^{ - 1}}(x))
Let tan⁡−1x=θ{\tan ^{ - 1}}x = \theta
Taking tan⁡−1{\tan ^{ - 1}} on the right-hand side we get
x=tan⁡θx = \tan \theta ----------- (1)
As we know that the sine can be written as the ratio of the tangent and secant
∴sin⁡θ=tan⁡θsec⁡θ\therefore \sin \theta = \dfrac{{\tan \theta }}{{\sec \theta }}-----------(2)
Since we know that identity of trigonometry which states that the square of secant is equal to the sum of square of tangent and one .i.e. sec⁡2θ=tan⁡2θ+1{\sec ^2}\theta = {\tan ^2}\theta + 1
We can say sec⁡θ=tan⁡2θ+1\sec \theta = \sqrt {{{\tan }^2}\theta + 1}
Now putting sec⁡θ=tan⁡2θ+1\sec \theta = \sqrt {{{\tan }^2}\theta + 1} in equation (2), we get
⇒sin⁡θ=tan⁡θtan⁡2θ+1\Rightarrow \sin \theta = \dfrac{{\tan \theta }}{{\sqrt {{{\tan }^2}\theta + 1} }}
Putting back the tan⁡θ=x\tan \theta = x in the RHS and θ=tan⁡−1x\theta = {\tan ^{ - 1}}x in LHS as we have assumed this in equation (1)
⇒sin⁡(tan⁡−1x)=xx2+1\Rightarrow \sin ({\tan ^{ - 1}}x) = \dfrac{x}{{\sqrt {{x^2} + 1} }}
Therefore, the simplification of expression sin⁡(tan⁡−1x)\sin ({\tan ^{ - 1}}x) is equal to xx2+1\dfrac{x}{{\sqrt {{x^2} + 1} }}

Note: In Mathematics the inverse trigonometric functions (every so often additionally called anti-trigonometric functions or cyclomatic function) are the reverse elements of the mathematical functions In particular, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are utilized to get a point from any of the point's mathematical proportions. Reverse trigonometric functions are generally utilized in designing, route, material science, and calculation.