![\bf [cot(\theta )+csc(\theta )]^2=\cfrac{1+cos(\theta )}{1-cos(\theta )} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the left-hand side}}{[cot(\theta )+csc(\theta )]^2}\implies cot^2(\theta )+2cot(\theta )csc(\theta )+csc^2(\theta ) \\\\\\ \cfrac{cos^2(\theta )}{sin^2(\theta )}+2\cdot \cfrac{cos(\theta )}{sin(\theta )}\cdot \cfrac{1}{sin(\theta )}+\cfrac{1}{sin^2(\theta )}\implies \cfrac{cos^2(\theta )}{sin^2(\theta )}+\cfrac{2cos(\theta )}{sin^2(\theta )}+\cfrac{1}{sin^2(\theta )}](https://tex.z-dn.net/?f=%5Cbf%20%5Bcot%28%5Ctheta%20%29%2Bcsc%28%5Ctheta%20%29%5D%5E2%3D%5Ccfrac%7B1%2Bcos%28%5Ctheta%20%29%7D%7B1-cos%28%5Ctheta%20%29%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Bdoing%20the%20left-hand%20side%7D%7D%7B%5Bcot%28%5Ctheta%20%29%2Bcsc%28%5Ctheta%20%29%5D%5E2%7D%5Cimplies%20cot%5E2%28%5Ctheta%20%29%2B2cot%28%5Ctheta%20%29csc%28%5Ctheta%20%29%2Bcsc%5E2%28%5Ctheta%20%29%20%5C%5C%5C%5C%5C%5C%20%5Ccfrac%7Bcos%5E2%28%5Ctheta%20%29%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%2B2%5Ccdot%20%5Ccfrac%7Bcos%28%5Ctheta%20%29%7D%7Bsin%28%5Ctheta%20%29%7D%5Ccdot%20%5Ccfrac%7B1%7D%7Bsin%28%5Ctheta%20%29%7D%2B%5Ccfrac%7B1%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%5Cimplies%20%5Ccfrac%7Bcos%5E2%28%5Ctheta%20%29%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%2B%5Ccfrac%7B2cos%28%5Ctheta%20%29%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%2B%5Ccfrac%7B1%7D%7Bsin%5E2%28%5Ctheta%20%29%7D)
![\bf \cfrac{\stackrel{\textit{perfect square trinomial}}{cos^2(\theta )+2cos(\theta )+1}}{sin^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the right-hand-side}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}}\implies \stackrel{\textit{multiplying by the denominator's conjugate}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}\cdot \cfrac{1+cos(\theta )}{1+cos(\theta )}}](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7B%5Cstackrel%7B%5Ctextit%7Bperfect%20square%20trinomial%7D%7D%7Bcos%5E2%28%5Ctheta%20%29%2B2cos%28%5Ctheta%20%29%2B1%7D%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%5Cimplies%20%5Cboxed%7B%5Ccfrac%7B%5Bcos%28%5Ctheta%20%29%2B1%5D%5E2%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Bdoing%20the%20right-hand-side%7D%7D%7B%5Ccfrac%7B1%2Bcos%28%5Ctheta%20%29%7D%7B1-cos%28%5Ctheta%20%29%7D%7D%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bmultiplying%20by%20the%20denominator%27s%20conjugate%7D%7D%7B%5Ccfrac%7B1%2Bcos%28%5Ctheta%20%29%7D%7B1-cos%28%5Ctheta%20%29%7D%5Ccdot%20%5Ccfrac%7B1%2Bcos%28%5Ctheta%20%29%7D%7B1%2Bcos%28%5Ctheta%20%29%7D%7D)
![\bf \cfrac{[1+cos(\theta )]^2}{\underset{\textit{difference of squares}}{[1-cos(\theta )][1+cos(\theta )]}}\implies \cfrac{[cos(\theta )+1]^2}{1^2-cos^2(\theta )} \\\\\\ \cfrac{[cos(\theta )+1]^2}{1-cos^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}}](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7B%5B1%2Bcos%28%5Ctheta%20%29%5D%5E2%7D%7B%5Cunderset%7B%5Ctextit%7Bdifference%20of%20squares%7D%7D%7B%5B1-cos%28%5Ctheta%20%29%5D%5B1%2Bcos%28%5Ctheta%20%29%5D%7D%7D%5Cimplies%20%5Ccfrac%7B%5Bcos%28%5Ctheta%20%29%2B1%5D%5E2%7D%7B1%5E2-cos%5E2%28%5Ctheta%20%29%7D%20%5C%5C%5C%5C%5C%5C%20%5Ccfrac%7B%5Bcos%28%5Ctheta%20%29%2B1%5D%5E2%7D%7B1-cos%5E2%28%5Ctheta%20%29%7D%5Cimplies%20%5Cboxed%7B%5Ccfrac%7B%5Bcos%28%5Ctheta%20%29%2B1%5D%5E2%7D%7Bsin%5E2%28%5Ctheta%20%29%7D%7D)
recall that sin²(θ) + cos²(θ) = 1, thus sin²(θ) = 1 - cos²(θ).
<u>Use a more precise conversion factor if you need a more precise measurement. 1 g = 0.00220462262 lb. 1 lb = 453.592 g (if you want to convert pounds to grams, multiply number of pounds by 453.592)</u>
Answer:
Doesn't have solution.
Step-by-step explanation:
x ≠ 0; x > 0 (I)
x - 5 > 0
x > 5 (II)
![log_6 x + log_6 (x + 5) = 2 \\\ log_6 [x(x + 5)] = 2 \\\ x^2 + 5x = 36 \\\ x^2 + 5x - 36 = 0 \\\ \Delta = 5^2 - 4.1.(- 36) \\\ \Delta = 169 \\\ \sqrt{\Delta} = \pm 13 \\\ x' = \frac{- 5 + 13}{2} \\\\ x' = 4 \\\ x" = \frac{- 5 - 13}{2} \\\\ x" = - 9](https://tex.z-dn.net/?f=log_6%20x%20%2B%20log_6%20%28x%20%2B%205%29%20%3D%202%20%5C%5C%5C%20log_6%20%5Bx%28x%20%2B%205%29%5D%20%3D%202%20%5C%5C%5C%20x%5E2%20%2B%205x%20%3D%2036%20%5C%5C%5C%20x%5E2%20%2B%205x%20-%2036%20%3D%200%20%5C%5C%5C%20%5CDelta%20%3D%205%5E2%20-%204.1.%28-%2036%29%20%5C%5C%5C%20%5CDelta%20%3D%20169%20%5C%5C%5C%20%5Csqrt%7B%5CDelta%7D%20%3D%20%5Cpm%2013%20%5C%5C%5C%20x%27%20%3D%20%5Cfrac%7B-%205%20%2B%2013%7D%7B2%7D%20%5C%5C%5C%5C%20x%27%20%3D%204%20%5C%5C%5C%20x%22%20%3D%20%5Cfrac%7B-%205%20-%2013%7D%7B2%7D%20%5C%5C%5C%5C%20x%22%20%3D%20-%209)
Then, this expression doesn't have solution.
I Hope I've helped you.