write the following functions in the simplest form :
1•tan-¹(cosx - sin x/cos x + sin x)
2• tan-¹(3a²x-x³/a³-3ax²)
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write the following functions in the simplest form :
1•tan-¹(cosx - sin x/cos x + sin x)
2• tan-¹(3a²x-x³/a³-3ax²)
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Question :-
Solution :-
→ tan-¹(cosx - sin x/cos x + sin x)
Divide Both Numerator & Denominator by cosx we get ,
→ tan-¹[{(cosx/cosx) - (sin x/cosx)} /{(cosx/cosx) + (sin x/cosx)}]
Using (sinx/cosx) = tanx now,
→ tan-¹[( 1 - tanx ) / (1 + tanx)]
Or,
→ tan-¹[ (1 - tanx) / (1 + 1 * tanx) ]
using 1 = tan(π/4) now,
→ tan-¹[ ( tan(π/4) - tanx) / (1 + tan(π/4)*tanx) ]
Comparing it with [ (tanA - tanB) / ( 1 + tanA*tanB) ]
→ tan-¹[ tan( π/4 - x ) ]
→ (π/4 - x) (Ans.)