We are working on verifing identities using trigonomic identities and such and of the about 50 or so problems we've had to complete i've been able to push through most except for these:
1.(COTx)(SECx)(SINx)=1
2.TANx + COTx = (SECx)(CSCx)
3.(COSt)(COTt) = 1-SIN^2t / SINt
Any help would be so greatly appreciated!!!
1. Write cot and sec in terms of sin and cos.
(cosx/sinx)(1/cosx)(sinx) = ?
See how the terms cancel?
2. (sinx/cosx) + (cosx/sinx)
Create a common denominagtr and add numerators.
= [sin^2x + cos^2x]/(sinx cosx)
= 1/(sinx cosx)
= (1/sinx)(1/cosx) = ?
3. I assume that there should be parentheses around (1-sin^2t). Replace it with cos^2 t and proceed as with the others.
Sure! I'd be happy to help you with these trigonometric identities.
Let's start with the first identity:
1. (COTx)(SECx)(SINx) = 1
To solve this, we'll use the fact that COTx = COSx/SINx and SECx = 1/COSx.
Substituting these values into the equation, we get:
(COSx/SINx)(1/COSx)(SINx) = 1
Canceling out the common terms of SINx and COSx, we have:
1 = 1
As you can see, the original equation holds true, so this identity is verified.
Moving on to the second identity:
2. TANx + COTx = (SECx)(CSCx)
We'll use the fact that TANx = SINx/COSx and COTx = COSx/SINx, as well as SECx = 1/COSx and CSCx = 1/SINx.
Substituting these values into the equation, we get:
(SINx/COSx) + (COSx/SINx) = (1/COSx)(1/SINx)
To simplify the left side of the equation, we need to find the common denominator.
Multiplying the first term by SINx/SINx and the second term by COSx/COSx, we get:
(SIN^2x/COSx + COS^2x/SINx) = (1/COSx)(1/SINx)
To simplify the left side further, we can use the identity SIN^2x + COS^2x = 1.
Substituting this identity into the equation, we have:
(1/COSx)(1/SINx) = (1/COSx)(1/SINx)
As you can see, the equation holds true, so this identity is also verified.
Finally, let's look at the third identity:
3. (COSt)(COTt) = (1 - SIN^2t)/SINt
We'll use the fact that COSt = 1/SINt and COTt = COSx/SINx, as well as SIN^2t + COS^2t = 1.
Substituting these values into the equation, we get:
(1/SINt)(COSx/SINx) = (1 - SIN^2t)/SINt
To simplify the left side of the equation, we can cross-cancel the common terms of SINt and SINx:
(1/COSx) = (1 - SIN^2t)/SINt
Now, let's simplify the right side by expanding the numerator:
(1/COSx) = (1 - SIN^2t)/SINt
(1/COSx) = (1 - SINt)(1 + SINt)/SINt
Canceling out the common terms of SINt in the numerator and denominator, we have:
(1/COSx) = (1 + SINt)
Rearranging the equation, we get:
1 = (1 + SINt)(COSx)
Expanding the equation further, we have:
1 = COSx + SINt * COSx
Finally, we can use the identity COSx = 1 - SIN^2x to substitute into the equation:
1 = (1 - SIN^2x) + SINt * (1 - SIN^2x)
1 = 1 - SIN^2x + SINt - SIN^2x * SINt
Simplifying, we have:
1 = 1 - SIN^2x + SINt - SIN^2t * SINx
At this point, it seems like there might be an error in the original equation or in our calculations. You may want to double-check the problem or your work to see if there are any mistakes.
I hope this explanation helps you understand the problem-solving process for these trigonometric identities! Let me know if you have any further questions.