Question

If secϴ = x + 1/4x, then prove that secϴ + tan ϴ = 2x or 1/2x.
(Class 10 Maths Sample Question Paper)

Answer 1

Given:

secϴ = x + 1/4x…….(1)

tan²ϴ = sec²ϴ -1
tan²ϴ =  (x + 1/4x)² -1  

[From equation 1]

tan²ϴ = x² + 1/16x² +½ -1
[ (a+b)² = a² + 2ab +b²]

tan²ϴ = x² + 1/16x² - ½
tan²ϴ = (x - 1/4x)²

[a² +b²-2ab = (a-b)²]
tanϴ = ±(x - 1/4x)

[Taking square roots both sides]
tanϴ = (x - 1/4x) or - (x - 1/4x)
When tanϴ =  (x - 1/4x), then  
secϴ +tanϴ = x +1/4x + x -1/4x = 2x

[From equation 1]

secϴ +tanϴ = 2x

When tanϴ = - (x - 1/4x), then  
secϴ +tanϴ = (x +1/4x) -( x -1/4x )  
 [From equation 1]

=  x +1/4x - x + 1/4x   
= 1/4x + 1/4x = 2/4x = 1/2x
secϴ +tanϴ = 1/2x

HOPE THIS WILL HELP YOU...

Answer 2

HELLO DEAR,


secθ = x + 1/4x ,

tan2 θ=sec2-1

=( x + 1/4x)2 - 1

=x2 + 1/16x2 + 2.x.1/4x -1

=x2 + 1/16x2 + 1/2 -1

= x2 + 1/16x2 -1/2



I HOPE ITS HELP YOU DEAR,
THANKS

= x2 + 1/16x2 -2.x.1/4x

=(x - 1/4x)2

tan θ = +(x - 1/4x) or - (x - 1/4x)

secθ + tanθ = = x + 1/4x +(x - 1/4x) 0r x + 1/4x - (x - 1/4x)

secθ + tanθ = 2x or 1/2x