Prove this identity
(cos è - 1) / (cos^2 è) =
sec è - sec^2 è
I have a question when you simplify the secant in terms of cosine. Would you cross multiply but then again there's the subtraction sign there instead of a multiplication sign? then would you multiply by the lcd to get the demoniator to be the same for both sides, im really confused, please help me, thanks
To prove this identity:
(cosθ - 1) / cos^2θ = secθ - sec^2θ
Let's start by simplifying both sides individually.
On the left side, we have: (cosθ - 1) / cos^2θ
To simplify this expression, we need to find a common denominator for (cosθ - 1) and cos^2θ. In this case, the common denominator is cos^2θ. Thus, we can rewrite the expression as:
(cosθ - 1) / cos^2θ = (cosθ - 1) * 1 / cos^2θ
Now, let's focus on the right side of the equation: secθ - sec^2θ.
The secant function can be expressed in terms of cosine as secθ = 1 / cosθ. Thus, we can rewrite the right side as:
1 / cosθ - (1 / cosθ)^2
To combine the terms, we need a common denominator. The common denominator is cosθ. Rewriting the expression, we get:
(1 - cosθ) / cosθ
Now, let's compare both sides of the equation:
(cosθ - 1) / cos^2θ = (1 - cosθ) / cosθ
To proceed with the proof, we can cross-multiply:
(cosθ - 1) * cosθ = cos^2θ * (1 - cosθ)
Expanding both sides:
cos^2θ - cosθ = cos^2θ - cos^3θ
Now, let's simplify further:
cos^2θ - cosθ = cos^2θ - cos^3θ
Rearranging terms:
0 = cos^3θ - cos^2θ + cosθ - cos^2θ
Simplifying:
0 = cos^3θ - 2cos^2θ + cosθ
We can see that both sides of the equation are equal to zero, indicating that the original equation is true.
Thus, we have proved the given identity:
(cosθ - 1) / cos^2θ = secθ - sec^2θ