Question

Simplify the following expression as much as possible.
csc^2 x * sin^2($\pi$/2 - x)/ cot^2 x

Answer 1

Answer: (1 - cos x)/cos^2 x

Step-by-step explanation: We can start by using the identity csc^2 x = 1/sin^2 x and cot^2 x = cos^2 x/sin^2 x to rewrite the expression as (1/sin^2 x) * sin^2(/2 - x) * (cos^2 x/sin^2 x)^-1. Then, we can simplify sin^2(/2 - x) to 1 - cos x and simplify the denominator to (1 - cos^2 x)/sin^2 x.

Finally, we can simplify the denominator to sin^2 x and cancel out the sin^2 x in the numerator and denominator, leaving us with (1 - cos x)/cos^2 x.

Answer 2

Answer:

$\dfrac{\csc^2 x \cdot \sin^2\left(\frac{\pi}{2}-x\right)}{\cot^2x}=1$

Step-by-step explanation:

Given rational trigonometric expression:

$\dfrac{\csc^2 x \cdot \sin^2\left(\frac{\pi}{2}-x\right)}{\cot^2x}$

Use the following trigonometric identities to rewrite the expression:

$\boxed{\begin{minipage}{5 cm}\underline{Trigonometric identities}\\\\ \cos x=\sin \left(\frac{\pi}{2}-x\right) \\\\ \csc x=\dfrac{1}{\sin x} \\\\\\ \cot x=\dfrac{\cos x}{\sin x} \\\end{minipage}}$

Therefore:

  $\dfrac{\csc^2 x \cdot \sin^2\left(\frac{\pi}{2}-x\right)}{\cot^2x}$

$=\dfrac{\left(\dfrac{1}{\sin x}\right)^2 \cdot (\cos x)^2}{\left(\dfrac{\cos x}{\sin x}\right)^2}$

$\textsf{Apply the exponent rule:} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}$

$=\dfrac{\dfrac{1}{\sin^2x} \cdot \cos^2x}{\dfrac{\cos^2x}{\sin^2x}}$

Simplify the numerator:

$=\dfrac{\dfrac{\cos^2x}{\sin^2x}}{\dfrac{\cos^2x}{\sin^2x}}$

$\textsf{Apply the fraction rule:}\;\; \dfrac{a}{a}=1$

$=1$

Therefore:

$\boxed{\dfrac{\csc^2 x \cdot \sin^2\left(\frac{\pi}{2}-x\right)}{\cot^2x}=1}$