Prove that
Answers
Explanation:
The given question involves trigonometric functions.
I have solved it using simple trigonometric conversions and manipulations.
I have also used some simple mathematical operations like addition, reciprocal and LCM etc.
I have given a detailed answer.
Kindly see the Attachment for the detailed answer.
I hope it will help you.
Question :- prove that sin 2x/sec( x+1) ×sec 2x/sec (2x+1) = tan(x/2) ?
Solution :-
solving LHS,
→ {sin2x/(secx+1)} * {sec2x/(sec2x+1)}
putting :-
- sec x = 1/cos x
- sec 2x = 1/cos 2x
→ {sin2x/(1/cosx)+1} * {(1/cos2x)/(1/cos2x)+1}
→ {2sinxcos²x/(1+cosx)} * {1/(1+cos2x)}
putting :-
- cos x = 2cos² x/2 - 1
- cos 2x = 2cos² x - 1
→ {2sinxcos²x/(1 + 2cos²x/2 - 1)} * {1/(1+2cos²x - 1)}
→ {2sinxcos²x/(2cos²x/2)} * {1/(2cos²x)}
→ sinx/(2cos²x/2)
putting :-
- sin x = 2 * sin x/2 * cos x /2
→ (2sinx/2cosx/2)/(2cos²x/2)
→ (sinx/2)/(cosx/2)
using :-
- sin x / cos x = tan x
→ (tanx/2) = RHS (Proved.)