if tanx + sinx is equal to m and 10 x - sinx is equal to n then m² - n²=
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Given that :
tan x+sin x=m. - (1)
tan x-sin x=n. - (2)
On square equation (1),
(tan x+sinx)² = m².
tan² x+sin² x+2tan x.sin x = m² - (3)
On square equation (2)
(tan x-sin x)² = n²
tan² x+sin² x-2tan x.sin x= n². - (4)
On substitute equation (3) and (4)
4tan x.sin x= m² - n². - (5)
On multiple equation (1) with equation (2)
(tan x+sin x)(tan x-sin x) = mn
[(a+b)(a-b) = a² - b²]
tan² x-sin² x = mn
(sin²xcos²x)−sin²x=mn
take sin² x as common
sin²x(1cos²x−1)=mn
sin²x(1−cos²x)cos²x=mn
tan² x.sin² x = mn
[sin² x/cos² x = tan² x]
[1-cos² x=sin² x]
Multiply by 16 both sides
16 tan² x.sin² x= mn
On taking square root
4tanx.sinx=4mn−−−√−(6)
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