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 November 10th, 2018, 11:33 PM #1 Newbie   Joined: Nov 2018 From: Norway Posts: 2 Thanks: 0 Finding exact value of the sum of two inverses Hi, From my textbook, I have an example saying: cos(arcsec(3) + arctan(2)) = (√5 − 4√10)/15 Why is it so? If anyone would want to explain, I would be very grateful. If this had been angles, I could recognize, ex: cos(arcsec(2) + arctan(1)) I would have used the sum identity for cos, so it would be cos(pi/3 + pi/4), It's especially this arcsec(3) which bothers me. Lars Last edited by skipjack; November 11th, 2018 at 03:08 AM. November 11th, 2018, 12:40 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,633 Thanks: 1472 \begin{align*} &\cos(\sec^{-1}(3)+\arctan(2)) = \\ &\cos(\sec^{-1}(3))\cos(\arctan(2)) - \sin(\sec^{-1}(3))\sin(\arctan(2)) = \\ &\dfrac 1 3 \dfrac{1}{\sqrt{5}} - \dfrac{2\sqrt{2}}{3}\dfrac{2}{\sqrt{5}} = \\ &\dfrac{1}{3\sqrt{5}}\left(1-4\sqrt{2}\right) = \\ &\dfrac{\sqrt{5}-4\sqrt{10}}{15} \end{align*} that $\cos(\sec^{-1}(3)) = \dfrac 1 3$ should be fairly obvious $\sin(\sec^{-1}(3)) = \dfrac{2\sqrt{2}}{3}$ comes from $\sin(x) = \sqrt{1-\cos^2(x)}$ to see $\cos(\arctan(2))$ and $\sin(\arctan(2))$ consider a right triangle with opposite side of length 2, adjacent side of length 1, and hypotenuse of length $\sqrt{5}$ Thanks from larsh November 11th, 2018, 04:38 AM #3 Newbie   Joined: Nov 2018 From: Norway Posts: 2 Thanks: 0 Oh, thanks. That made a lot of sense I think I sometimes don't see the forest for all the trees. It totally made sense and with your guidance I was able to reproduce the solution and understand everything. One million thanks's!  https://image.ibb.co/mgBk3V/20181111-133353.jpg Tags exact, finding, inverses, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post alishab Trigonometry 8 December 11th, 2015 07:31 PM pwndogmaster1 Trigonometry 1 December 1st, 2014 08:17 PM hirano Algebra 5 September 12th, 2010 05:56 AM mtt0216 Algebra 16 March 8th, 2010 05:09 PM kmjt Algebra 1 December 6th, 2009 05:15 AM

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