I hope this is in the right place.

Prove the following statement by a direct proof of the contrapositive statement.
If two lines are perpendicular, the product of their gradients is -1.
The contrapositive would be:
If the products of two gradients is not -1, then their lines are not perpendicular.
My problem is I don't know how to go about proving the statement.
I tried this:
$\tan (\theta-\phi)=\frac{\tan\theta-\tan\phi}{1-\tan\theta\tan\phi}$
$\tan\theta\tan\phi=1-\frac{\tan\theta-\tan\phi}{\tan (\theta-\phi)}$
$\frac{\tan\theta-\tan\phi}{\tan (\theta-\phi)}$ is not equal to 2
But I don't know how to continue.
Thanks

2. tan(θ - φ) = (tanθ - tanφ)/(1 + tanθ*tanφ)

3. Oops my bad. that would make it
$\tan\theta\tan\phi=\frac{\tan\theta-\tan\phi}{\tan (\theta-\phi)}-1$?
What's after this? How am I supposed to prove that if the product of two gradients is not -1, then the lines are not perpendicular? It would mean $\frac{\tan\theta-\tan\phi}{\tan (\theta-\phi)}$ is not 0.

4. If the two lines are perpendicular to each other, then (θ - φ) = π/2 and tan(π/2) = infinity. It is possible only when (1 + tanθ*tanφ) = 0 or tanθ*tanφ = -1.

5. I don't think I understand what you mean. I'm supposed to prove the statement from its contrapositive statement. That is I'm supposed to prove that if the product of the gradients of two lines is not -1, then the lines are not perpendicular.

6. Originally Posted by arze
I don't think I understand what you mean. I'm supposed to prove the statement from its contrapositive statement. That is I'm supposed to prove that if the product of the gradients of two lines is not -1, then the lines are not perpendicular.
If tanθ*tanφ is not equal to -1, 1+tanθ*tanφ is not equal to zero and tan(θ - φ) is not equal to infinity. That means (θ - φ) is either more than π/2 or less than π/2. Hence the two lines are not perpendicular to each other.