The coefficient of the highest power of $x$ in the expansion of $\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8$ is

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Aicharya · Accepted Answer

$\begin{aligned} & 	ext {Since }\left(x+\sqrt{x^2-1}ight)^8+\left(x-\sqrt{x^2-1}ight)^8 \ & =2\left\{\begin{array}{r} { }^8 C_0 x^8+{ }^8 C_2 x^6\left(x^2-1ight)+{ }^8 C_4 x^4\left(x^2-1ight)^2 \ \left.+{ }^8 C_6 x^2\left(x^2-1ight)^3+{ }^8 C_8 x^0\left(x^2-1ight)^4ight\} \end{array}ight. \end{aligned}$ So coefficient of highest power of $x$ $\begin{aligned} & =2\left\{{ }^8 C_0+{ }^8 C_2+{ }^8 C_4+{ }^8 C_6+{ }^8 C_8ight\} \ & =(1+1)^8+(1-1)^8=2^8=256 \end{aligned}$

The coefficient of the highest power of $x$ in the expansion of…

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