I don't understand how to simplify this. Please help!
(1-2cos^2ө)/ (sin ө-cosө) + cotө
To simplify the expression (1-2cos^2ө)/(sin ө-cosө) + cotө, we can use trigonometric identities to rewrite the terms in a simpler form. Here's a step-by-step guide:
Step 1: Rewrite cos^2ө as (cos ө)^2:
(1-2(cos ө)^2)/(sin ө - cos ө) + cot ө
Step 2: Apply the Pythagorean identity sin^2ө + cos^2ө = 1, and solve for sin^2ө:
sin^2ө = 1 - cos^2ө
Step 3: Substitute sin^2ө with 1 - cos^2ө:
(1-2(cos ө)^2) / ((1 - cos^2 ө) - cos ө) + cot ө
Step 4: Simplify the denominator by combining like terms:
(1-2(cos ө)^2) / (1 - cos^2 ө - cos ө) + cot ө
Step 5: Apply the distributive property of multiplication to separate the terms in the denominator:
(1-2(cos ө)^2) / (1 - cos ө - cos^2 ө) + cot ө
Step 6: Rearrange the terms in the denominator:
(1-2(cos ө)^2) / (-cos^2 ө - cos ө + 1) + cot ө
Step 7: Factor out a negative sign from the denominator:
(1-2(cos ө)^2) / (-(cos^2 ө + cos ө - 1)) + cot ө
Step 8: Factorize the quadratic expression in the denominator:
(1-2(cos ө)^2) / (-(cos ө - 1)(cos ө + 1)) + cot ө
Now, the expression is simplified as much as possible.