Answer:
(-2/21)√21
Explanation:
You want the exact value of tan(arccsc(-5/2)).
Equivalent
We know that csc(x) = 1/sin(x), so the angle that has a csc of -5/2 will have a sin of -2/5. This means the expression can be written as ...
tan(arcsin(-2/5))
Further, we can use tan=sin/cos to rewrite this as ...
tan(arccsc(-5/2)) = sin(arcsin(-2/5))/cos(arcsin(-2/5))
= (-2/5)/cos(arcsin(-2/5))
The cosine is found from the identity ...
cos(x) = √(1 -sin(x)²)
cos(arcsin(-2/5)) = √(1 -(-2/5)²) = (√21)/5
Simplified
Using this value in the above expression, we get ...
tan(arccsc(-5/2)) = (-2/5)/((√21)/5) = -2/√21
tan(arccsc(-5/2)) = (-2/21)√21 . . . . . . . . . . rationalize the denominator
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Additional comment
A good calculator can provide a head start. This one (in the first attachment) likes to leave the radical in the denominator, as it gives a more compact expression.
The second attachment shows the square of the answer using the cosecant function. This is to demonstrate the equivalence we claimed above.
The exact values of these things are often irrational square roots, so we checked that possibility by squaring the decimal result. That gave a fraction that tells us the exact value is -√(4/21) = (-2/21)√21.