write each expression in terms of sine and cosine:
csc^3 theta/cot theta * tan^2 theta/sec^2 theta
To write the given expression in terms of sine and cosine, we need to express each trigonometric function using sine and cosine.
First, let's rewrite the trigonometric functions in terms of sine and cosine:
csc(theta) is equal to 1/sin(theta)
cot(theta) is equal to cos(theta)/sin(theta)
tan(theta) is equal to sin(theta)/cos(theta)
sec(theta) is equal to 1/cos(theta)
Using these conversions, let's rewrite the expression:
csc^3(theta)/cot(theta) * tan^2(theta)/sec^2(theta)
= (1/sin^3(theta))/(cos(theta)/sin(theta)) * (sin^2(theta)/cos^2(theta))/(1/cos^2(theta))
= (1/sin^3(theta)) * (sin^2(theta)/cos(theta)) * (cos^2(theta)) * (sin(theta))
Now, cancel out similar terms:
= 1 * sin^2(theta) * 1 * sin(theta)
= sin^3(theta) * sin(theta)
= sin^4(theta)
Therefore, the expression csc^3(theta)/cot(theta) * tan^2(theta)/sec^2(theta) can be written as sin^4(theta).