Hi. Can you verify the following:
(cosθ) / (sinθ – 1) + (cosθ)/ (1 + sinθ)
(1 / cos^2θ) – (2) (1/cos^2θ) (sinθ / cosθ) + (sin^2θ / cos^2θ)
I see no equal signs
What are we verifying ?
Oooops. Its this one. Sorry.
(cosθ) / (sinθ – 1) + (cosθ)/ (1 + sinθ) = -2tanθ
(1 / cos^2θ) – (2) (1/cos^2θ) (sinθ / cosθ) + (sin^2θ / cos^2θ) = (1 - sinθ) / (1 + sinθ)
To verify the expressions given, we need to manipulate them to simplify and determine if they are equal. Let's start with the first expression:
(cosθ) / (sinθ – 1) + (cosθ) / (1 + sinθ)
To simplify this expression, we can use a common denominator. The common denominator would be (sinθ – 1)(1 + sinθ). Let's calculate:
= [(cosθ)(1 + sinθ) + (cosθ)(sinθ – 1)] / [(sinθ – 1)(1 + sinθ)]
Expanding the terms within the brackets:
= [cosθ + cosθsinθ + cosθsinθ – cosθ] / [(sinθ – 1)(1 + sinθ)]
Combine like terms:
= [2cosθsinθ] / [(sinθ – 1)(1 + sinθ)]
Now, let's simplify the second expression:
(1 / cos^2θ) – (2) (1/cos^2θ) (sinθ / cosθ) + (sin^2θ / cos^2θ)
Start by simplifying the second term by combining fractions with the same denominator:
= (1 / cos^2θ) – (2sinθ / cos^3θ) + (sin^2θ / cos^2θ)
Finding the common denominator for all three terms, which would be cos^3θ, we obtain:
= (cos^3θ + (-2sinθ) + (sin^2θ)) / cos^3θ
Combining the terms within the numerator:
= (cos^3θ – 2sinθ + sin^2θ) / cos^3θ
Now, compare the two expressions:
[(2cosθsinθ) / ((sinθ – 1)(1 + sinθ))] vs. [(cos^3θ – 2sinθ + sin^2θ) / cos^3θ]
The two expressions are not equal, so we can conclude that the given expressions are not equivalent.