Question

If $\cos ec\theta + \cot \theta = m$ and $\cos ec\theta - \cot \theta = n$ , prove that mn=1.

Answer 1

Hint: Here we have to simplify the equations using trigonometry identities to find the product of both equations.

Complete step-by-step answer:
Given $\cos ec\theta + \cot \theta = m$ and $\cos ec\theta - \cot \theta = n$
To prove: mn=1
Taking LHS, $mn = \left( {\cos ec\theta + \cot \theta } \right)\left( {\cos ec\theta - \cot \theta } \right)$
Using formula $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
$\Rightarrow$ $mn = \cos e{c^2}\theta - {\cot ^2}\theta $
We know that $\cos e{c^2}\theta = 1 + {\cot ^2}\theta \Rightarrow \cos e{c^2}\theta - {\cot ^2}\theta = 1$
$ \Rightarrow $ mn=1=RHS
Hence Proved.

Note: These types of problems can be easily solved with the help of understanding of trigonometric identities and formulas.