Question

If sin(a+b) /cos(a-b) =1-x/1+x then show that tan(tan45-a) (tan45-b) =x

Answer 1

Answer:

If sin(a+b) /cos(a-b) =1-x/1+x then  (tan45-a) (tan45-b) =x

Step-by-step explanation:

Sin(a+b)/Cos(a-b) = (1-x)/ (1+x)

Tan(45-a) . Tan(45-b) = x

Tan (45-a) = (Tan 45 - Tan a)/(1 + Tan 45. Tan a ) =  (1 - Tan a)/(1 + Tan a)

= ( 1- Sin a /Cos a) / ( 1 + Sin a / Cos a)  =  (Cos a - Sin a) / (Cos a + Sin a)

Similarly Tan (45-b) = (Cos b - Sin b) / (Cos b + Sin b)

=> x =  ((Cos a - Sin a) / (Cos a + Sin a))((Cos b - Sin b) / (Cos b + Sin b))

=> x = ((cos a cos b + Sin a Sin b) - (Sin a Cos b + cos a Sin b) ) / ((cos a cos b + Sin a Sin b) + (Sin a Cos b + cos a Sin b) )

Using Cos(a-b) = Cos a Cos b + Sin a Sin b

Sin(a +b) = Sin a Cos b + Cos a Sin b

=> x = (Cos (a-b) - Sin(a+b) )/ (Cos (a-b) + Sin(a+b))

1-x = 2Sin(a+b) /(Cos (a-b) + Sin(a+b))

1 + x = 2 Cos(a+b) /(Cos (a-b) + Sin(a+b))

using above two

1-x/(1+x) = Sin(a+b)/Cos(a-b)