tanA=ntanB
sinA=msinB
find cos^2A in terms of 'm' and 'n'
plz help
taking reciprocals, we have
cot^2A = 1/n^2 cot^2B
csc^2A = 1/m^2 csc^2B
m^2 csc^2A - n^2 cot^2A = csc^2B-cot^2B = 1
Now you should be able to get things down to cos^2A
To find cos^2A in terms of 'm' and 'n', we can use the Pythagorean Identity: sin^2A + cos^2A = 1.
Given that sinA = m * sinB, we can write sin^2A = (m * sinB)^2 = m^2 * sin^2B.
Similarly, given that tanA = n * tanB, we know that tanA = sinA / cosA and tanB = sinB / cosB. This implies that sinA = n * sinB * cosB. Squaring both sides, we get sin^2A = n^2 * sin^2B * cos^2B.
Now, substitute these results into the Pythagorean Identity:
m^2 * sin^2B + cos^2A = 1
Since sin^2A = m^2 * sin^2B and sin^2A = n^2 * sin^2B * cos^2B, we can substitute these as follows:
n^2 * sin^2B * cos^2B + cos^2A = 1
Next, rearrange the equation to solve for cos^2A:
cos^2A = 1 - n^2 * sin^2B * cos^2B
cos^2A = 1 - n^2 * (1 - cos^2A) * cos^2B
cos^2A = 1 - n^2 * cos^2B + n^2 * cos^2A * cos^2B
Simplifying further, we have:
cos^2A - n^2 * cos^2B * cos^2A = 1 - n^2 * cos^2B
cos^2A (1 - n^2 * cos^2B) = 1 - n^2 * cos^2B
cos^2A = (1 - n^2 * cos^2B) / (1 - n^2 * cos^2B)
Therefore, cos^2A in terms of 'm' and 'n' is given by:
cos^2A = (1 - n^2 * cos^2B) / (1 - n^2 * cos^2B)
To find cos^2A in terms of 'm' and 'n', we can use the trigonometric identity:
sin^2A + cos^2A = 1
From the given information, we have:
tanA = n * tanB
sinA = m * sinB
We can rewrite these equations using sine and cosine:
sinA = (n * sinB) / cosB
sinA = m * sinB
Since sinA is common in both equations, we can equate the two expressions:
(n * sinB) / cosB = m * sinB
Now, we can solve for cosB by dividing both sides of the equation by sinB:
(n * cosB) = m
Dividing both sides by n, we get:
cosB = m/n
Using the Pythagorean identity, we know that sin^2B + cos^2B = 1:
sin^2B + (m/n)^2 = 1
Substituting cos^2B = (m/n)^2 into the identity, we get:
sin^2B + cos^2B = 1
sin^2B + cos^2A = 1
Therefore, cos^2A in terms of 'm' and 'n' is:
cos^2A = 1 - sin^2A
cos^2A = 1 - (m * sinB)^2
cos^2A = 1 - m^2 * sin^2B
Note: This is the general form of cos^2A in terms of 'm' and 'n' based on the given equations.