UPSC NDA Math Model Paper 3

Given: cot A ⋅ cot B = 2
⇒cos(A+B)‌sec(A−B)=

cos(A+B)

cos(A−B)

As we know that, cos(A+B)=cos‌A‌cos‌B−sin‌ ‌B and cos(A−B)=cos‌A‌cos‌B+sin‌A‌sin‌B
⇒cos(A+B)‌sec(A−B) =

cos‌A‌cos‌B−sin‌A‌sin‌B

cos‌A‌cos‌B+sin‌A‌sin‌B

Now by dividing both the numerator and denominator by sin A sin B, we get
⇒cos(A+B)‌sec(A−B) =

cot‌A‌cot‌B−1

cot‌A‌cot‌B+1

By, substituting the value of cot A ⋅ cot B in the above equation, we get
⇒cos(A+B)‌sec(A−B) =

2−1

2+1

=

1

3