Is simplifying a complete expansion of the right hand side of a trigonometric identity sufficient to prove it?

My first question is, when proving a trigonometric identity for all real values of a variable such as $ x$ , is it sufficient to expand the right hand side into elementary functions and simplify until it equals the left hand side?

I know this is very basic but since I have seen complicated proofs of identities by induction, I want to be sure that this is a sufficient method of proof if the problem allows. My second question is, is this considered a direct proof?

My third question is, assuming that the answers to my first two questions are affirmative, would there be any reason that a trigonometric identity need be proven any way other than the way I specified in my first question? If trigonometric expressions can be rewritten in terms of the elementary trig functions, would there ever be need for an induction proof rather than direct?

Thank you in advance.

What’s the point of a hand crossbow?

I have seen the following information a few different places, and it raises the question for me:

If the hand crossbow requires two hands to use–one for shooting, one for loading–then is there any benefit to using it over a two-handed crossbow?

You still need a free hand, so you can’t use a shield or a second weapon, and you can always hold a two-handed crossbow in one hand while you grab something else or cast a spell.

Ammunition and Two-Handed errata

A couple of things that have basically been how I’ve been playing things, but good to see them officially in the rules. You need a hand free to reload a one-handed weapon with the Ammunition property. (It needs two hands). Also, although the maul and other two-handed weapons might require two hands to attack with, you can hold those weapons in a single hand.

This second clarification makes a spell-caster casting Somatic spells have an easier time with it when also wielding a two-handed weapon.

launch daemon that will not run at boot, but will run later when loaded by hand

I have a launchd daemon definition /Library/LaunchDaemons/homebrew.mxcl.docker-machine.plist which contains:

<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"> <plist version="1.0">   <dict>     <key>EnvironmentVariables</key>     <dict>         <key>PATH</key>         <string>/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/bin</string>     </dict>     <key>Label</key>     <string>homebrew.mxcl.docker-machine</string>     <key>ProgramArguments</key>     <array>         <string>/usr/local/opt/docker-machine/bin/docker-machine</string>         <string>start</string>         <string>default</string>     </array>     <key>RunAtLoad</key>     <true/>     <key>KeepAlive</key>     <true/>     <key>UserName</key>     <string>sysbh</string>     <key>WorkingDirectory</key>     <string>/Users/sysbh</string>   </dict> </plist> 

At boot time, this does not work. But when logged in, I can get it to run with sudo launchctl load /Library/LaunchDaemons/homebrew.mxcl.docker-machine.plist

Any tips what I can do to find out why the launchd plist doesn’t get the program started during boot/startup, but does later?

And suppose (which is a distinct possibility) that this daemon depends on another launchd daemon having been run first (and to make it complicated: that daemon only runs once to install kexts), is there a way around that problem?

What qualifies as a “light personal hand baggage carried inside passenger compartment” exempt from the 6 HKD charge when taking a taxi in Hong Kong?

https://www.td.gov.hk/en/transport_in_hong_kong/public_transport/taxi/taxi_fare_of_hong_kong/index.html (mirror) indicates that “Every piece of baggage (Except light personal hand baggage carried inside passenger compartment)” is charged 6 HKD charge when taking a taxi in Hong Kong.

What qualifies as a “light personal hand baggage carried inside passenger compartment”? E.g., does the regulation give any concrete numbers of the maximum dimension or weight for a baggage to qualify as a light personal hand baggage?


https://www.td.gov.hk/en/transport_in_hong_kong/public_transport/taxi/taxi_fare_of_hong_kong/index.html (mirror):

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Wielding a Hand Crossbow and a Rapier with Crossbow Expert Feat while dropping & picking up the Rapier

I am trying to see if I am breaking any rules of D&D 5e here.

I am a Swashbuckler Rogue (possibly irrelevant).

I am wielding a hand crossbow and a rapier in each hand, with the Crossbow Expert feat (PHB, p165). On my turn I use my action to hit with my rapier, then drop the rapier as a free action, shoot the hand crossbow as bonus action, then pick up rapier as my Object Interaction. Аm I breaking any game rules here or am I good?

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