If cosec x - secx = √5 then find cosecx + secx
Answers
Given : cosec x - secx = √5
To find : cosecx + secx
Solution:
cosec x - secx = √5
Let say
cosecx + secx = y
Adding both
2Cosecx = √5 + y
=> 2/Sinx = √5 + y
=> Sinx = 2/(y + √5)
2Secx = y - √5
=> 2/Cosx = y - √5
=> Cosx = 2/(y - √5)
Sin²x + Cos²x = 1
=> (2/(y + √5))² + (2/(y - √5))² = 1
=> 4((y - √5)² + (y + √5)² ) = (y - √5)² (y + √5)²
=> 4 ( y² + 5 -2√5y + y² + 5 + 2√5y) = (y² - 5)²
=> 8( y² + 5) = (y² - 5)²
=> y⁴ -18y² - 15 = 0
=> y² = (18 ± √(-18)² - 4(1)(-15) )/2
=> y² = (18 ± 8√6) )/2
=> y² = 9 ± 4√6
=> y² = 9 + 4√6 ( as Square can not be - ve)
=> y = √( 9 + 4√6 )
=> y = 4.3357
cosecx + secx = √( 9 + 4√6 ) = 4.3357
x = 17.718°
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