Prove that: $sin ,cot^-1 cos , an^-1x = sqrtdfracx^2+1x^2+2 $.

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This question was taken from Miscellaneous Example of S.L. Loney"s Trigonometry.

Since, the question involves 2 parts, LHS và RHS, LHS is completely Trigonometric while, RHS is algebraic, so my approach was to get taylor series for $x+1$ then differentiate it và keep on doing to lớn show that both sides are same... But, I wonder if there is an trigonometric solution for the question too or not!

Please help.


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When you compose trig functions with inverse trig you get algebraic functions. I always draw a right triangle. If we first assume the angle is in the first quadrant, we have
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where $ an^-1x$ is angle $A$ & $cos A=frac 1sqrt1+x^2$. Now draw a new triangle for the next two functions. Then you need to lớn make sure it works for other quadrants because of the ranges of the inverse trig functions.


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Let $y= an^-1x$. Then $ an y=x$ and so drawing a right triangle with angle $y$ and sides $1,x,sqrt1+x^2$, we find $cos y=dfrac1sqrt1+x^2$.

Now let $t=cot^-1(cos y)$. Then $cot t=cos y=dfrac1sqrt1+x^2$, so drawing a right triangle with angle $t$, we find the hypotenuse is $sqrt2+x^2$, so $sin t=sqrtdfrac1+x^22+x^2$.


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