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Question: If \(af\left( \tan x \right)+bf\left( \cot x \right)=x,\) then \({{f}^{-1}}\left( \cot x \right)=\) ...

If af(tanx)+bf(cotx)=x,af\left( \tan x \right)+bf\left( \cot x \right)=x, then f1(cotx)={{f}^{-1}}\left( \cot x \right)=
1)1ab\left. 1 \right) \dfrac{1}{a-b}
2)sin2xa+b\left. 2 \right) \dfrac{{{\sin }^{2}}x}{a+b}
3)sin2xab\left. 3 \right) \dfrac{{{\sin }^{2}}x}{a-b}
4)sin2xba\left. 4 \right) \dfrac{{{\sin }^{2}}x}{b-a}

Explanation

Solution

We should remember some of the trigonometric identities for solving this problem. We know that the first derivative of the Tangent function is the square of the Secant function. Similarly, the first derivative of the Cotangent function is the negative of the square of the Cosecant function.

Complete step by step solution:
Let us consider the given function, af(tanx)+bf(cotx)=x.......(1)af\left( \tan x \right)+bf\left( \cot x \right)=x.......\left( 1 \right)
We are going to find the first derivative of the whole equation given. For that, we need to differentiate the equation once with respect to the variable x.x.
We are going to use the derivatives of trigonometric functions, ddxtanx=sec2x\dfrac{d}{dx}\tan x={{\sec }^{2}}x and ddxcotx=cosec2x.\dfrac{d}{dx}\cot x=-\cos e{{c}^{2}}x.
Also, we need to remember the facts: when we differentiate f(tanx),f\left( \tan x \right), we first differentiate the function and then differentiate the Tangent function. Similarly, when we differentiate f(cotx),f\left( \cot x \right), we first differentiate the function and then differentiate the Cotangent function.
So, we will get ddxaf(tanx)=addxf(tanx)=af(tanx)ddxtanx=af(tanx)(sec2x)\dfrac{d}{dx}af\left( \tan x \right)=a\dfrac{d}{dx}f\left( \tan x \right)=a{f}'\left( \tan x \right)\dfrac{d}{dx}\tan x=a{f}'\left( \tan x \right)\left( {{\sec }^{2}}x \right)
and ddxaf(cotx)=addxf(cotx)=af(cotx)ddxcotx=af(cotx)(cosec2x).\dfrac{d}{dx}af\left( \cot x \right)=a\dfrac{d}{dx}f\left( \cot x \right)=a{f}'\left( \cot x \right)\dfrac{d}{dx}\cot x=a{f}'\left( \cot x \right)\left( -\cos e{{c}^{2}}x \right).
Now we will get ddx(af(tanx)+bf(cotx))=dxdx.\dfrac{d}{dx}\left( af\left( \tan x \right)+bf\left( \cot x \right) \right)=\dfrac{dx}{dx}.
Since dxdx=1,\dfrac{dx}{dx}=1, we will get ddxaf(tanx)+ddxbf(cotx)=1.\dfrac{d}{dx}af\left( \tan x \right)+\dfrac{d}{dx}bf\left( \cot x \right)=1.
We have applied the linearity property of differentiation ddx(au+bv)=adudx+bdvdx\dfrac{d}{dx}\left( au+bv \right)=a\dfrac{du}{dx}+b\dfrac{dv}{dx} in the above step.
We can again use the linearity property to get addxf(tanx)+bddxf(cotx)=1.a\dfrac{d}{dx}f\left( \tan x \right)+b\dfrac{d}{dx}f\left( \cot x \right)=1.
Now, we can substitute the derivatives of the functions we have found above in the equation obtained, af(tanx)sec2x+bf(cotx)(cosec2x)=1.a{f}'\left( \tan x \right){{\sec }^{2}}x+b{f}'\left( \cot x \right)\left( -\cos e{{c}^{2}}x \right)=1.
Let us transpose the second summand from the LHS to the RHS, we will get
af(tanx)sec2x=1bf(cotx)(cosec2x)=1+bf(cotx)(cosec2x).\Rightarrow a{f}'\left( \tan x \right){{\sec }^{2}}x=1-b{f}'\left( \cot x \right)\left( -\cos e{{c}^{2}}x \right)=1+b{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right).
Now let us transpose asec2xa{{\sec }^{2}}x from the LHS to the RHS,
f(tanx)=1+bf(cotx)(cosec2x)asec2x.......(2)\Rightarrow {f}'\left( \tan x \right)=\dfrac{1+b{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)}{a{{\sec }^{2}}x}.......\left( 2 \right)
Now, put x=π2xx=\dfrac{\pi }{2}-x in the equation (1).\left( 1 \right).
We will get af(tan(π2x))+bf(cot(π2x))=π2x.af\left( \tan \left( \dfrac{\pi }{2}-x \right) \right)+bf\left( \cot \left( \dfrac{\pi }{2}-x \right) \right)=\dfrac{\pi }{2}-x.
Now we will get af(cotx)+bf(tanx)=π2x.af\left( \cot x \right)+bf\left( \tan x \right)=\dfrac{\pi }{2}-x.
We are going to differentiate the above equation with respect to x.x.
We will get, ddx(af(cotx)+bf(tanx))=ddx(π2x).\dfrac{d}{dx}\left( af\left( \cot x \right)+bf\left( \tan x \right) \right)=\dfrac{d}{dx}\left( \dfrac{\pi }{2}-x \right).
Since ddx(π2x)=1,\dfrac{d}{dx}\left( \dfrac{\pi }{2}-x \right)=-1, we will get af(cotx)(cosec2x)+bf(tanx)sec2x=1.a{f}'\left( \cot x \right)\left( -\cos e{{c}^{2}}x \right)+b{f}'\left( \tan x \right){{\sec }^{2}}x=-1.
Now let us transpose the first summand from the LHS to the RHS, we will get
bf(tanx)sec2x=1af(cotx)(cosec2x)=af(cotx)(cosec2x)1.\Rightarrow b{f}'\left( \tan x \right){{\sec }^{2}}x=-1-a{f}'\left( \cot x \right)\left( -\cos e{{c}^{2}}x \right)=a{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-1.
Now let us transpose bsec2xb{{\sec }^{2}}x from the LHS to the RHS,
f(tanx)=af(cotx)(cosec2x)1bsec2x.......(3)\Rightarrow {f}'\left( \tan x \right)=\dfrac{a{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-1}{b{{\sec }^{2}}x}.......\left( 3 \right)
Now, from the equations (2)\left( 2 \right) and (3)\left( 3 \right) we will get,
f(tanx)=1+bf(cotx)(cosec2x)asec2x=af(cotx)(cosec2x)1bsec2x.\Rightarrow {f}'\left( \tan x \right)=\dfrac{1+b{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)}{a{{\sec }^{2}}x}=\dfrac{a{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-1}{b{{\sec }^{2}}x}.
Let us cancel sec2x{{\sec }^{2}}x from both the fractions,
1+bf(cotx)(cosec2x)a=af(cotx)(cosec2x)1b.\Rightarrow \dfrac{1+b{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)}{a}=\dfrac{a{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-1}{b}.
Transpose aa and bb to get,
a2f(cotx)(cosec2x)a=b+b2f(cotx)(cosec2x).\Rightarrow {{a}^{2}}{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-a=b+{{b}^{2}}{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right).
Now we will transpose b+b2f(cotx)(cosec2x)b+{{b}^{2}}{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right) from the RHS to the LHS and aa from the LHS and the RHS,
a2f(cotx)(cosec2x)b2f(cotx)(cosec2x)=a+b.\Rightarrow {{a}^{2}}{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)-{{b}^{2}}{f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)=a+b.
Now let us take the common factor out,
(a2b2)f(cotx)(cosec2x)=a+b.\Rightarrow \left( {{a}^{2}}-{{b}^{2}} \right){f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)=a+b.
We transpose a2b2{{a}^{2}}-{{b}^{2}} from the LHS to the RHS,
f(cotx)(cosec2x)=a+b(a2b2).\Rightarrow {f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)=\dfrac{a+b}{\left( {{a}^{2}}-{{b}^{2}} \right)}.
Now we will get,
f(cotx)(cosec2x)=a+b(a+b)(ab).\Rightarrow {f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)=\dfrac{a+b}{\left( a+b \right)\left( a-b \right)}.
We cancel (a+b)\left( a+b \right) from the numerator and the denominator,
f(cotx)(cosec2x)=1(ab).\Rightarrow {f}'\left( \cot x \right)\left( \cos e{{c}^{2}}x \right)=\dfrac{1}{\left( a-b \right)}.
We transpose cosec2x\cos e{{c}^{2}}x from the LHS to the RHS,
f(cotx)=1(ab)(cosec2x).\Rightarrow {f}'\left( \cot x \right)=\dfrac{1}{\left( a-b \right)\left( \cos e{{c}^{2}}x \right)}.
Since (sin2x)=1(cosec2x),\left( {{\sin }^{2}}x \right)=\dfrac{1}{\left( \cos e{{c}^{2}}x \right)}, we will get f(cotx)=sin2x(ab).{f}'\left( \cot x \right)=\dfrac{{{\sin }^{2}}x}{\left( a-b \right)}.
Hence f(cotx)=sin2x(ab).{f}'\left( \cot x \right)=\dfrac{{{\sin }^{2}}x}{\left( a-b \right)}.
So, the correct answer is “Option 3”.

Note: We know that tan(π2x)=cotx,cot(π2x)=tanx,sec(π2x)=cosecx\tan \left( \dfrac{\pi }{2}-x \right)=\cot x, \cot \left( \dfrac{\pi }{2}-x \right)=\tan x, \sec \left( \dfrac{\pi }{2}-x \right)=\cos ecx and coesc(π2x)=secx.coesc\left( \dfrac{\pi }{2}-x \right)=\sec x. Since all the trigonometric functions are positive in the first quadrant, the sign will not change in any of the above identities. Also, a2b2=(a+b)(ab).{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right).