Prove the following identities:
1. (tan theta - sin theta)^2 + (1-cos theta)^2 = (1-sec theta) ^2
2. (1-2cos^2 theta) / sin theta cos theta = tan theta - cot theta
3. (sin theta + cos theta ) ^2 + (sin theta - cos theta ) ^2 = 2
Thank you so much! :)
sinθ = tanθ cosθ, so
tanθ - sinθ = tanθ (1-cosθ)
squared that is tan^2θ (1-cosθ)^2
now we can add up the left side:
tan^2θ(1-cosθ)^2 + (1-cosθ)^2
= (tan^2θ + 1)(1-cosθ)^2
= sec^2θ (1-cosθ)^2
on the right side, we have
1-secθ = (cosθ-1)/cosθ
squared, that is sec^2θ (cosθ-1)^2
and they are the same
You're welcome! I'll explain how to prove each of these identities.
1. (tan theta - sin theta)^2 + (1-cos theta)^2 = (1-sec theta) ^2
To prove this identity, we will expand both sides of the equation. Let's start with the left side:
(tan theta - sin theta)^2 + (1-cos theta)^2
= (tan^2 theta - 2tan theta sin theta + sin^2 theta) + (1 - 2cos theta + cos^2 theta)
= tan^2 theta - 2tan theta sin theta + sin^2 theta + 1 - 2cos theta + cos^2 theta
Now let's simplify the right side of the equation:
(1-sec theta)^2
= (1 - 1/cos theta)^2
= (1 - 2/cos theta + 1/cos^2 theta)
= 1 - 2/cos theta + 1/cos^2 theta
Now we can see that both sides of the equation are equal. Therefore, the identity is proved.
2. (1-2cos^2 theta) / sin theta cos theta = tan theta - cot theta
To prove this identity, we will start with the left side of the equation:
(1-2cos^2 theta) / sin theta cos theta
= (1 - 2(cos^2 theta)) / (sin theta cos theta)
= (1 - 2(1 - sin^2 theta)) / (sin theta cos theta)
= (1 - 2 + 2sin^2 theta) / (sin theta cos theta)
= (1 + 2sin^2 theta) / (sin theta cos theta)
Now let's simplify the right side of the equation:
tan theta - cot theta
= sin theta / cos theta - cos theta / sin theta
= (sin^2 theta - cos^2 theta) / (sin theta cos theta)
= (1 - cos^2 theta - cos^2 theta) / (sin theta cos theta)
= (1 - 2cos^2 theta) / (sin theta cos theta)
Again, both sides of the equation are equal. Therefore, the identity is proved.
3. (sin theta + cos theta ) ^2 + (sin theta - cos theta ) ^2 = 2
To prove this identity, we will expand both sides of the equation. Let's start with the left side:
(sin theta + cos theta ) ^2 + (sin theta - cos theta ) ^2
= (sin^2 theta + 2sin theta cos theta + cos^2 theta) + (sin^2 theta - 2sin theta cos theta + cos^2 theta)
= 2sin^2 theta + 2cos^2 theta
Now we can simplify the right side:
2
= sin^2 theta + cos^2 theta
Again, both sides of the equation are equal. Therefore, the identity is proved.
I hope this helps! Let me know if you have any more questions.