[Sin^2(theta) - tan(theta) ] / [cos^2(theta) - cot(theta)] = tan^2(theta)
I am supposed to prove that the left side of the equation is supposed to be equal to the right side. And I been confused on how to solve it.
As a general rule, I change all trig ratios to sines or cosines unless I recognize
a definite identity, so ...
LS = (sin^2 θ - sinθ/cosθ) / (cos^2 θ - cosθ/sinθ)
= [ (cosθsin^2 θ - sinθ)/cosθ ] / [ (sinθcos^2 θ - cosθ)/sinθ ]
= (cosθsin^2 θ - sinθ)/cosθ * sinθ /(sinθcos^2 θ - cosθ)
= sinθ(sinθcosθ - 1)/cosθ * sinθ/(cosθ(sinθcosθ - 1) )
= sin^2 θ / cos^2 θ
= tan^2 θ
= RS
To prove that the left side of the equation ([sin^2(theta) - tan(theta)] / [cos^2(theta) - cot(theta)]) is equal to the right side (tan^2(theta)), we need to simplify the expression on the left side and show that it simplifies to the expression on the right side.
Let's start by simplifying the expression on the left side step-by-step:
1. Start with the numerator: [sin^2(theta) - tan(theta)]. We can write tan(theta) as sin(theta)/cos(theta), so the numerator becomes:
[sin^2(theta) - sin(theta)/cos(theta)].
2. To combine the terms in the numerator, we need a common denominator, which is cos(theta). So, the numerator becomes:
[(sin^2(theta) * cos(theta))/cos(theta) - (sin(theta)/cos(theta))].
3. Simplify the numerator:
[(sin^2(theta) * cos(theta) - sin(theta))/cos(theta)].
4. Now, let's simplify the denominator: [cos^2(theta) - cot(theta)]. We can write cot(theta) as cos(theta)/sin(theta), so the denominator becomes:
[cos^2(theta) - cos(theta)/sin(theta)].
5. To combine the terms in the denominator, we need a common denominator, which is sin(theta). So, the denominator becomes:
[(cos^2(theta) * sin(theta))/sin(theta) - (cos(theta)/sin(theta))].
6. Simplify the denominator:
[(cos^2(theta) * sin(theta) - cos(theta))/sin(theta)].
Now, we can rewrite the original expression with the simplified numerator and denominator:
[(sin^2(theta) * cos(theta) - sin(theta))/cos(theta)] / [(cos^2(theta) * sin(theta) - cos(theta))/sin(theta)].
To simplify this fraction, we can multiply the numerator by the reciprocal of the denominator:
[(sin^2(theta) * cos(theta) - sin(theta))/cos(theta)] * [sin(theta)/(cos^2(theta) * sin(theta) - cos(theta))].
Next, simplify further by canceling out common factors:
[(sin^2(theta) * cos(theta) - sin(theta))/cos(theta)] * [sin(theta)/(cos(theta) * sin(theta) - cos(theta))].
[(sin^2(theta) * cos(theta) - sin(theta))/(cos(theta) * sin(theta) - cos(theta))].
Now, factor out sin(theta) from the numerator and the denominator:
[sin(theta) * (sin(theta) * cos(theta) - 1)] / [cos(theta) * (sin(theta) - 1)].
Notice that sin(theta)/cos(theta) is equal to tan(theta), so we can rewrite the expression as:
[sin(theta) * tan(theta)] / [cos(theta) * (sin(theta) - 1)].
Finally, we can simplify further by recognizing that sin(theta) / cos(theta) is equal to tan(theta):
tan(theta) * [sin(theta) / (cos(theta) * (sin(theta) - 1))].
Now, we have tan(theta) multiplied by the fraction [sin(theta) / (cos(theta) * (sin(theta) - 1))].
To prove that this expression is equal to tan^2(theta), we can rewrite tan^2(theta) as sin^2(theta) / cos^2(theta):
sin^2(theta) / cos^2(theta).
So, we have shown that the left side of the equation ([sin^2(theta) - tan(theta)] / [cos^2(theta) - cot(theta)]) simplifies to the right side (tan^2(theta)).
By simplifying both sides of the equation, we have demonstrated that they are equivalent.