Solution
Step 1
Write the trigonometric equation
[tex]tan^2\theta\text{ - sin}^2\theta\text{ = sin}^4\theta sec^2\theta[/tex]
Step 2
Apply the quotient identity
[tex]\frac{sin^2\theta}{cos^2\theta}\text{ - sin}^2\theta[/tex]
Step 3
Factor out the common factor
[tex]sin^2\theta(\frac{1}{cos^2\theta}-\text{ 1\rparen}[/tex]
Step 4
simplify
[tex]sin^2\theta(\frac{1-cos^2\theta}{cos^2\theta})[/tex]
Step 5
Reciprocal identity
[tex]sin^2\theta(1\text{ - cos}^2\theta)sec^2\theta[/tex]
Step 6
Use the Pythagorean identity
[tex]\begin{gathered} sin^2\theta\text{ + cos}^2\theta\text{ = 1} \\ sin^2\theta\text{ = 1 - cos}^2\theta \\ Hence \\ sin^2\theta(1\text{ - cos}^2\theta)sec^2\theta \\ sin^2\theta\text{ }\times\text{ sin}^2\theta\text{ }\times\text{ sec}^2\theta \end{gathered}[/tex]
Step 7
Simplify
[tex]sin^4\theta sec^2\theta[/tex]
Final answer
Option D
IV , I , II , V , III , IV , II
