Prove: csc &#x2061;<!-- ⁡ --> a + cot &#x2061;<!-- ⁡ --> a = cot &#x2061;<!-- ⁡

lnwlf1728112xo85f 2022-05-07 Answered
Prove: csc a + cot a = cot a 2
All I have right now, from trig identities, is
1 sin a + 1 tan a = 1 tan ( a / 2 )
Where do I go from there?
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Answers (2)

glapaso7ng5
Answered 2022-05-08 Author has 13 answers
We start with the following identities: sin ( 2 a ) = 2 sin a cos a cos ( 2 a ) = 1 2 sin 2 a
We solve these to get the half-angle identities: sin ( a ) = 2 sin a 2 cos a 2 sin 2 a 2 = 1 2 ( 1 cos a )
We now tackle the problem
1 sin a + cos a sin a = cos a 2 sin a 2
multiplying out both sides we get that
2 sin 2 a 2 ( cos a + 1 ) = 2 sin a 2 cos a 2 sin a
Using the identities above we get that
( 1 cos a ) ( 1 + cos a ) = sin 2 a
1 cos 2 a = sin 2 a
1 = sin 2 a + cos 2 a
We now have a trivial trigonometric identity, so the equivalence is proved
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poklanima5lqp3
Answered 2022-05-09 Author has 5 answers
csc a + cot a = cot a 2
L.H.S.
1 sin a + cos a sin a 1 + cos a sin a 1 + 2 cos 2 a 2 1 sin a 2 cos 2 a 2 2 sin a 2 cos a 2 cot a 2 = R.H.S
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