Question

If `13sectheta - 5tantheta =13` then the value of `13tantheta - 5sectheta` can be?

Answer 1

`pm5`

Explanation:

`{(a/costheta-bsintheta/costheta=a),(-b/costheta+asintheta/costheta=x):}`

after multiplying the first equation by `b` and the second by `a` with posterior add term to term we have

`(a^2-b^2)sintheta/costheta=a b + a x`

after multiplying the first equation by `a` and the second by `b` with posterior add term to term we have

`(a^2-b^2)/costheta=a^2+bx` then

`a b+ax=(a^2+bx)sintheta` so

`{(sintheta=(a(b+x))/(a^2+bx)),(costheta=(a^2-b^2)/(a^2+bx)):}`

but `sin^2theta+cos^2theta=1`

then

`a^2(b+x)^2+(a^2-b^2)^2=(a^2+bx)^2`

solving for `x` we obtain

`x=pmb = pm5`

Answer 2

Given

`13sectheta-5tantheta=13`

`=>(13sectheta-5tantheta)^2=13^2`

`=>13^2sec^2theta+5^2tan^2theta-2*13*5secthetatantheta=13^2`

`=>13^2(sec^2theta-1)+5^2sec^2theta-5^2-2*13*5secthetatantheta=0`

`=>13^2tan^2theta+5^2sec^2theta-2*(13tantheta)*(5sectheta)=5^2`

`=>(13tantheta-5sectheta)^2=5^2`

`=>(13tantheta-5sectheta)=pm5`