To rewrite the expression tan 0 x csc 0 in terms of sine and cosine, we can use the following trigonometric identities:
The reciprocal identity: csc 0 = 1/sin 0
The tangent identity: tan 0 = sin 0/cos 0
The Pythagorean identity: sin^2 0 + cos^2 0 = 1
Substituting these identities into the given expression, we get:
tan 0. csc 0 = (sin 0/cos 0) * (1/sin 0)
Using the reciprocal identity and the Pythagorean identity, we can simplify this expression further to get:
tan 0. csc 0 = (sin 0 * 1)/(cos 0 * sin 0)
Next, we can cancel out the sin 0 term on the numerator and denominator to get:
tan 0. csc 0 = 1/cos 0
Finally, we can use the Pythagorean identity again to rewrite the denominator as:
tan 0. csc 0 = 1/sqrt(1 - sin^2 0)
Therefore, the final simplified expression is:
tan 0. csc 0 = 1/sqrt(1 - sin^2 0)
This is the final answer.