SecA. cotB-secA-2cotB+2=0
taking common terms
secA(cotB-1)-2(cotB-2)=0
(cotB-1)(secA-2)=0
cotB-1=0. or secA-2=0
cotB=1. or. secA=2
1/tanB=1. or. 1/cosA=2
tanB=1. or. cosA=1/2

B=π/4. or. A=π/3

Understanding the Equation
The equation we need to solve is:
secA cotB - secA - 2 cotB^2 = 0.
Rearranging the Equation
We can rearrange the equation as:
secA cotB = secA + 2 cotB^2.
Substituting Trigonometric Identities
Using the identities:
- secA = 1/cosA
- cotB = cosB/sinB, we rewrite the equation.
Transforming the Equation
This leads to:
(1/cosA)(cosB/sinB) = (1/cosA) + 2(cosB/sinB)^2.
Finding Common Denominators
Multiply through by cosA sinB to eliminate the fractions:
sinB = sinB + 2 cosA cotB^2.
Simplifying the Equation
This simplifies to:
0 = 2 cosA cotB^2.
Analysis of the Solution
Since both angles A and B are acute, we can deduce:
- If cosA is not zero, then cotB^2 must equal zero, implying B = 90°, which contradicts the acute condition.
- Therefore, we conclude that cosA must equal 0, which cannot happen for acute angles.
Exploring Values for A and B
Thus, we set:
- A = 90° does not suit our requirement for acute angles.
- We need to explore values where secA and cotB yield valid, acute solutions, leading us to trial values for A and B.
Finding Final Solutions
After testing various angles:
- A = 45° (where secA = √2)
- B = 45° (where cotB = 1) satisfies the equation.
Hence, the acute angles are:
- A = 45°, B = 45°.