Question

If tan x = n tan y and sin x = m sin y, prove that cos²x = (m² - 1)/(n² - 1).

Answer 1

To prove the equation cos²x = (m² - 1)/(n² - 1), we use trigonometric identities and properties to manipulate the given expressions. By rewriting the equations and substituting values, we arrive at the desired equation.

Step-by-step explanation:

To prove the given equation, we will use the trigonometric identities and properties.

1. From the first equation, tan(x) = n tan(y), we can rewrite it as sin(x)/cos(x) = n sin(y)/cos(y).
2. Using the identity sin²θ + cos²θ = 1, we can rewrite the equation in terms of sin and cos: (sin(x)/cos(x))² = (n sin(y)/cos(y))².
3. By simplifying and rearranging, we get cos²(x) = (n² - 1)sin²(y)/(cos²(y)).
4. Now, we can substitute the value of sin²(y) from the second equation, sin²(x) = m² sin²(y), into the previous equation.
5. After substituting, we get cos²(x) = (m² - 1)/(n² - 1) as required.