sinA+cosA=p ,secA+cosecA=q (cosA)^2/[2-2(sinA)^2-(cosecA)^2]=p^2/(2(p^2)-q^2
please complete the full equation
Let's solve the given equation step by step.
First, let's rearrange the given equations:
sinA + cosA = p (1)
secA + cosecA = q (2)
Now, let's simplify the right-hand side of the equation:
(cosA)^2 / [2 - 2(sinA)^2 - (cosecA)^2] = p^2 / [2(p^2) - q^2]
Using the identity cos^2A = 1 - sin^2A and secA = 1/cosA, we can rewrite the equation as:
cos^2A / [2 - 2sin^2A - (1/sin^2A)] = p^2 / [2p^2 - q^2]
Next, simplify the denominator by combining like terms:
cos^2A / [(2sin^2A - 1/sin^2A) - 2] = p^2 / [2p^2 - q^2]
To simplify this further, let's write everything in terms of sinA:
cos^2A = 1 - sin^2A
1/sin^2A = csc^2A
Substituting these values and simplifying the equation, we get:
(1 - sin^2A) / [(2sin^2A - csc^2A) - 2] = p^2 / [2p^2 - q^2]
Multiplying both sides of the equation by (2sin^2A - csc^2A - 2), we have:
1 - sin^2A = p^2 * (2sin^2A - csc^2A - 2) / [2p^2 - q^2]
Expanding the numerator on the right-hand side, we get:
1 - sin^2A = (2p^2sin^2A - p^2csc^2A - 2p^2) / [2p^2 - q^2]
Multiplying both sides by (2p^2 - q^2), we have:
(2p^2 - q^2) - (2p^2sin^2A - p^2csc^2A - 2p^2)sin^2A = 0
Simplifying this equation further:
2p^2 - q^2 - 2p^2sin^2A + p^2csc^2A * sin^2A - 2p^2sin^2A = 0
2p^2 - q^2 - 4p^2sin^2A + p^2csc^2A * sin^2A = 0
Rearranging the equation:
p^2csc^2A * sin^2A - 4p^2sin^2A - q^2 + 2p^2 = 0
Now you have the final equation.
To solve this equation, we will first simplify both sides separately and then equate them.
Given equations:
1. sinA + cosA = p
2. secA + cosecA = q
We'll start with the left-hand side (LHS) of the equation:
(cosA)^2 / [2 - 2(sinA)^2 - (cosecA)^2]
To simplify this, we'll convert sinA to cosA using the Pythagorean identity sin^2A = 1 - cos^2A:
(cosA)^2 / [2 - 2(1 - cos^2A) - (1 / sin^2A)]
Simplifying further:
(cosA)^2 / [2 - 2 + 2(cos^2A) - 1 / sin^2A]
(cosA)^2 / [2(cos^2A) + 1 / sin^2A]
Using the reciprocal identity (1 / sin^2A = cosec^2A):
(cosA)^2 / [2(cos^2A) + cosec^2A]
Now let's simplify the right-hand side (RHS) of the equation:
p^2 / [2(p^2) - q^2]
Now we can equate the two sides:
(cosA)^2 / [2(cos^2A) + cosec^2A] = p^2 / [2(p^2) - q^2]
At this point, we have transformed the given equation into an expression involving trigonometric functions. In order to solve for A, we need additional equations or constraints to determine the values of p and q. Without any additional information, it is not possible to find a unique solution for A.
If there are any additional constraints or equations, please provide them, and I'll be happy to help you solve the problem.
you don't offer any ideas at all, but maybe this will get you started
p^2 = (sinA+cosA)^2 = sin^2A+2sinAcosA+cos^2A = 1+2sinAcosA
q^2 = (1/cosA+1/sinA)^2 = (sin^2A+cos^2A)/(sin^2Acos^2A) = 1/(sin^2A cos^2A)
cos^2A/[2-2(sinA)^2-(cosecA)^2]
= cos^2A/(2cos^2A - 1/sin^2A)
= (sin^2A cos^2A)/(2sin^2A cos^2A - 1)
See where you can go with that.