Question

Q. if asecx - btanx = 1 and a²sec²x - b²tan²x = 5 , then (4a² - 9b²) = ?

Answer 1

a²sec²x -b²tan²x =5
= (asecx+btanx)(asecx-btanx) =5
= (asecx+btanx)*1=5
= asecx+btanx=5. (I)
Given, asecx-btanx=1. (ll)
Adding (l) and (ll), we get
2asecx=6
=a = 3/secx
Similarly, we get,
b= 2/tanx

So,
4a²=4* 9/sec²x
=36/sec²x

9b²=9*4/tan²x
=36/tan²x

4a²-9b²= 36/sec²x-36/tan²x
= 36( 1/sec²x - 1/tan²x)
= 36( tan²x-sec²x/sec²x tan²x)
=36(-(sec²x-tan²x)/sec²xtan²x)
= 36(-1/sec²xtan²x)
= 36(-cos²xcot²x)
= -36cos²xcot²x