Question

tan theta/ tan (90°- theta) + sin (90°- theta)/ cos theta = sec2 theta

Answer 1

Answer:

Step-by-step explanation:GIVEN EQUATION : acosx + bsinx = c

Provided roots of the equation - α and β

HENCE , acosα + bsinα = c , acosβ + bsinβ = c

Subtracting the subsequent second equation from the first , we obtain ;

a(cosα - cosβ) + b(sinα - sinβ) = 0

or ,

b ( sinα - sinβ ) - a ( cosβ - cosα ) = 0

or ,

2b cos α + β/2 sin α - β/2 = 2asin α + β/2 sin α- β /2

or ,

Tan α+ β /2 =b/a [ because α , β are different angles , hence can be substituted as sinα-β/2≠0

Hence , Sin (α+β)= 2Tan α+β/2 / 1 + Tan² α+β /2

= 2 { b/a} / 1 + b²/a² = 2ab / a² + b²

Hence , by further reductions ,

Tan (α+β)=2Tan(α+β/2)/1-Tan²( α+β /2 ) = 2ab / a² - b²

HENCE PROVED