Question

The angle a lies between 0° and 90° and is such that

2 tan²a + sec²a = 5-4 tana

Show that

3 tan²a +4 tana -4 = 0
and hence find the exact value of tan a

Answer 1

Good.
Now we know this property
Sec^2 A = 1 + tan^2 A
Now lets replace the sec square term with this in the given question
We get,
3tan^2(A) + 1 = 5 - 4tanA
Hence 3tan^2A + 4tanA - 4 =0
Hence proved
Lets take tanA as a value x.
Thus 3x2+4x-4=0
=> 3x2 + 6x - 2x - 4 =0
=> 3x(x+2) -2(x+2) =0
=> (3x-2)(x+2)=0
x=2/3 or x=-2
But tanA cant have a negative value as 0Thus tanA=x= 2/3