How do I prove the identities of these questions? 1+tan2u1−tan2u=1cos2u−sin2u sin⁡x+sin⁡(5x)cos⁡x+cos⁡(5x)=tan⁡(3x)

trefoniu1

trefoniu1

Answered

2022-01-28

How do I prove the identities of these questions?
1+tan2u1tan2u=1cos2usin2u
sinx+sin(5x)cosx+cos(5x)=tan(3x)

Answer & Explanation

ul2ph3ojc

ul2ph3ojc

Expert

2022-01-29Added 12 answers

You will need two facts, first the definition of tangent, second main trigonometric identity:
tanx=sinxcosx
sin2x+cos2x=1
Using that you may transform (1) as:
1+tan2u1tan2u=1+sin2ucos2u1sin2ucos2u=
=cos2u+sin2ucos2ucos2usin2ucos2u=1cos2ucos2ucos2usin2u=1cos2usin2u
The original (2) which is:
sinx+sin5xcosx+cos5x=tan3x
is false as shown in another answer (take x=π4 for instance). Here is a visualization of LHS and RHS

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