make the indicated trigonometric substitution in the given algebraic expression and simplify. assume that (…

make the indicated trigonometric substitution in the given algebraic expression and simplify. assume that ( 0 < \theta < pi / 2 ).\n( sqrt { x ^ { 2 } - 1 }, quad x = csc ( \theta ) )

make the indicated trigonometric substitution in the given algebraic expression and simplify. assume that ( 0 < \theta < pi / 2 ).\n( sqrt { x ^ { 2 } - 1 }, quad x = csc ( \theta ) )

Answer

Explanation:

Step1: Substitute (x = \csc(\theta))

Substitute (x=\csc(\theta)) into (\sqrt{x^{2}-1}), we get (\sqrt{\csc^{2}(\theta)-1}).

Step2: Use trigonometric identity

Recall the trigonometric identity (1+\cot^{2}(\theta)=\csc^{2}(\theta)), so (\csc^{2}(\theta)-1 = \cot^{2}(\theta)). Then (\sqrt{\csc^{2}(\theta)-1}=\sqrt{\cot^{2}(\theta)}).

Step3: Simplify the square - root

Since (0 < \theta<\frac{\pi}{2}), (\cot(\theta)>0). So (\sqrt{\cot^{2}(\theta)}=\cot(\theta)).

Answer:

(\cot(\theta))