Question

tan theta1=k cot theta2 then cos (theta1-theta2)÷cos (theta1-theta 2)​

Answer 1

Given:

tan theta1= k cot theta2  

To find:

Then the value of cos (theta1 - theta2) ÷ cos (theta1 - theta 2)​ ?

 

Solution:

From given, we have,

tan theta1= k cot theta2  

we know the relation between tan and cot,

tan = 1/cot

so, we have,

tan theta1= k [ 1/tan theta2 ]

→ tan theta1 × tan theta 2 = k

Now consider the given,  

cos (theta1 - theta2) ÷ cos (theta1 + theta 2)​

we use the formulae,

cos (a - b) = cos a cos b - sin a sin b

cos (a + b) = cos a cos b + sin a sin b

cos (theta1 - theta2) ÷ cos (theta1 + theta 2)​  

= (cos theta1 cos theta 2 - sin theta1 sin theta2) ÷ (cos theta1 cos theta2 - sin theta1 sin theta2)

we know the relation between sin, cos and tan,

tan = sin / cos

dividing both numerator and denominator by cos theta1 cos theta 2, we get,

→ cos (theta1 - theta2) ÷ cos (theta1 + theta 2)​  = [1 - tan theta1 tan theta2] / [1 + tan theta1 tan theta2]  

as we have already calculated the value of tan theta1 × tan theta 2 = k, upon substituting we get,

∴ cos (theta1 - theta2) ÷ cos (theta1 + theta 2)​  =  (1 - k)/(1 + k)