CosA x cos2A + ((sin2A)^2)/2cosA
To simplify the given expression, we will start by using trigonometric identities to rewrite it in a more manageable form. Let's break it down step by step:
1. Start with the given expression: cosA x cos2A + ((sin2A)^2)/2cosA
2. Apply the double-angle identity for cosine: cos2A = 2(cos^2A) - 1
3. Substitute the value of cos2A from step 2 into the expression: cosA x [2(cos^2A) - 1] + ((sin2A)^2)/2cosA
4. Distribute the cosA term: 2cosA(cos^2A) - cosA + ((sin2A)^2)/2cosA
5. Simplify the expression within parentheses: 2cos^3A - cosA + ((sin2A)^2)/2cosA
6. Apply the double-angle identity for sine: sin2A = 2sinAcosA
7. Square the sin2A term: ((2sinAcosA)^2)/2cosA
8. Simplify the expression: 2cos^3A - cosA + (4sin^2Acos^2A)/2cosA
9. Cancel out the 2 in the denominator: 2cos^3A - cosA + 2sin^2Acos^2A/cosA
10. Further simplify the last term by canceling cosA: 2cos^3A - cosA + 2sin^2AcosA
So, the simplified expression is 2cos^3A - cosA + 2sin^2AcosA.