Question

evaluate sec^2 theta-(sin^2 theta -2sin^4theta / 2cos^4theta - cos^2 theta ) =1 ​

Answer 1

Given

sec²∅ - (sin²∅ - 2sin⁴∅)/(2cos⁴∅ - cos²∅) = 1

LHS:

sec²∅ - (sin²∅ - 2sin⁴∅)/2cos⁴∅ - cos²∅)

= sec²∅ - [sin²∅(1 - 2sin²∅)]/[cos²∅(2cos²∅ - 1)]

= sec²∅ - tan²∅(1-2sin²∅)/(2cos²∅-1)

We know that,

cos2∅ = 1 - 2sin²∅ = 2cos²∅ - 1

Thus,

sec²∅ - tan²∅(cos2∅/cos2∅)

= sec²∅ - tan²∅

= 1

Hence,proved