Question

Cos x/cos(x-2y)=lambda then tan(x-y)tany

Answer 1

Given: cos x / cos ( x - 2y ) = lambda

To find: The value of tan(x-y)tany

Solution:

• So here we have given cos x/ cos ( x - 2y ) = lambda

• Now tan ( x - y ) tan y can be written as:

                tan ( x - y ) tan y = (sin ( x - y ) / cos ( x - y )) x (sin y / cos y)

• Now multiply and divide by 2, we get:

                2/2 x  (sin ( x - y ) / cos ( x - y )) x (sin y / cos y)

• Now we know:

                2sinasinb  cos(a-b) - cos(a+b) and 2cosacob = cos(a-b) + cos(a+b)

• Applying this, we get:

                cos(x - y - y ) - cos (x - y + y) / cos(x - y - y) + cos(x - y + y)

                cos(x - 2y ) - cos (x ) / cos(x - 2y) + cos(x)

• Now taking cos(x - 2y ) common, we get:

                cos(x - 2y ) { 1 - cos (x )/cos(x - 2y ) } / cos(x - 2y ) { 1 + cos (x )/cos(x - 2y ) }

• Cancelling cos(x - 2y ), we get:

                 { 1 - cos (x )/cos(x - 2y ) } / { 1 + cos (x )/cos(x - 2y ) }

• Now we have given cos x / cos ( x - 2y ) = lambda, so :

                {1 - lambda / 1 + lambda }

Answer:

         So the value of tan ( x - y ) tan y is {1 - lambda / 1 + lambda }