Question

If $\sin x+\csc x=2$, then ${\sin }^{3n}x+{\csc }^{3n}x$ equals

• A. $2^{2n-1}$
• B. $2^{2n}$
• C. $2^{3n+1}$
• D. $2$

Answer 1

Answer: (D)

$2$

$\sin x+\csc x=2$

${\sin }^{3n}x+{\csc }^{3n}x=?$

squaring both sides

$\Rightarrow {\sin }^{2}x+{\csc }^{2}x+2=4$

$\Rightarrow {\sin }^{2}x+{\csc }^{2}x=2\Rightarrow {\sin }^{n}x+{\csc }^{n}x=2$

Now, $({\sin }^{n}x{)}^3+({\csc }^{n}x{)}^3=({\sin }^{n}x+{\csc }^{n}x)\left(\right({\sin }^{n}x{)}^2+({\csc }^{n}x{)}^2-{\sin }^{n}x{\csc }^{n}x)$

$=2(2-1)=2$

$\therefore$ The result is $2$.