How to simplify this trigonometric equation using identities?

Question

The answer is tan(10) if that helps at all, and the directions say to simplify this using appropriate identities! (NOTE: The apostrophes do not denote derivatives... they're just there to keep everything aligned.)

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cot 5(1 - tan²5)

What baffles me is the random cot stuck in there. It's not attached to anything, there's no variable after it, etc. I transcribed it just how it is in the book. I don't know if it means the cot of 5(1 - tan²5) or what, but it's driving me crazy!

I'll pick a best answer TODAY if anyone can help me with this! Please explain what steps you're using if at all possible! Thanks!

Answer 1

Since cot 5 = 1/tan 5, this can be rewritten as:

2 tan(5) / (1-tan^2 (5))

and recall the tangent double angle formula:

tan(2x) = 2 tan(x) / (1-tan^2(x))

Replace x with 5, and you have:

tan(10) = 2 tan(5) / (1-tan^2(5))

which is true.

Answer 2

= 2 tan 5 / [ (1 - tan ² 5) ]
= 2 tan 5 / [( 1 - tan 5) (1 + tan 5) ]
= 2 tan 5 / [1 - tan ² 5 ]
= tan (5 + 5)
= tan 10

Answer 3

to start up, i could artwork interior the parenthesis and convert sec to (a million/cos) and then come across a difficulty-loose denominator. Sorry, i don't do the homework for faculty teenagers. i'm a math instructor...and inspire you to crack your e book open, and sweat, bleed, and curse slightly with the intention to do the mathematics, and get the mathematics. we are able to do the mathematics for you, yet you will not learn something. hint on the 2d 2: in basic terms look at a graph and see the place the sin and tan purposes circulate the x-axis. this is as ordinary as that! the two those have countless answer factors, so which you will could desire to point a development to your instructor.

Answer 4

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cot 5(1 - tan²5)
= 2/[ cot 5 - cot 5 tan²5]
= 2/[ 1/tan5 - tan²5/tan5]
= 2/ [ 1/tan5 - tan5]
= 2/[ (1 - tan²5) /tan5]
= 2 tan5 / (1 - tan²5)
= tan 2A
etc