Proving Identities:
2 columns
(tan + cot)^2 = sec^2 + csc^2
I'm having trouble breaking down the left side to = the right side..
Any help please
left
(sin/cos + cos/sin)^2
sin^2/cos^2 + 2 + cos^2/sin^2
[sin^4 +2sin^2 cos^2+cos^4 ]/cos^2 sin^2
(sin^2+cos^2)^2/cos^2sin^2
1^2/sin^2cos^2
1/sin^2 cos^2
right
1/cos^2 + 1/sin^2
sin ^2/cos^2sin^2 + cos^2/cos^2 sin^2
1/cos^2 sin^2
Hi Damon .. apparently they want the right side to stay "as is" and for the left side to transform into exactly what the right side says .... sorry
I do not think so. That would be a very unusual thing for "them" to say :)
The question says: Set up a 2 column proof to show that each of the following equations is an identity. Transform the left side to become the right side.
(tan + cot)^2 = sec^2 + csc^2
(tan + cot)^2 = tan^2 + 1 + cot^2
= sec^2 - 1 + 2 + csc^2 - 1
= sec^2 + csc^2
oops that's tan^2 + 2 + cot^2
To prove the identity (tan + cot)^2 = sec^2 + csc^2, first, let's simplify the left side step by step:
(tan + cot)^2
We can expand this expression using the binomial square formula, which states that (a + b)^2 = a^2 + 2ab + b^2.
Substituting a = tan and b = cot, we have:
(tan + cot)^2 = tan^2 + 2(tan)(cot) + cot^2
Next, we can simplify tan and cot in terms of sine and cosine using the definitions:
tan = sin/cos
cot = cos/sin
Substituting these values, the expression becomes:
(tan + cot)^2 = (sin/cos)^2 + 2(sin/cos)(cos/sin) + (cos/sin)^2
Simplifying further, we get:
(tan + cot)^2 = sin^2/cos^2 + 2(sin/cos)(cos/sin) + cos^2/sin^2
Next, let's simplify the middle term of the expression:
2(sin/cos)(cos/sin)
= 2((sin * cos) / (cos * sin))
= 2/1
= 2
Replacing this value in the expression, we have:
(tan + cot)^2 = sin^2/cos^2 + 2 + cos^2/sin^2
Since sin^2/cos^2 is equal to 1/tan^2 and cos^2/sin^2 is equal to 1/cot^2, we can rewrite the expression as follows:
(tan + cot)^2 = 1/tan^2 + 2 + 1/cot^2
Now, we can simplify 1/tan^2 and 1/cot^2 as follows:
1/tan^2 = sec^2
1/cot^2 = csc^2
Substituting these values, the expression becomes:
(tan + cot)^2 = sec^2 + 2 + csc^2
And there you have it, we have proved the identity (tan + cot)^2 = sec^2 + csc^2.