Math, asked by manjitlal244, 4 months ago

prove that :cos7x+cos5x/sin7x+sin5x=cotx​

Answers

Answered by anindyaadhikari13
14

Required Answer:-

Given to prove:

 \rm \mapsto \dfrac{ \cos(7x) +  \cos(5x) }{ \sin(7x) -  \sin(5x)  }  =  \cot(x)

Proof:

We know that,

 \rm \mapsto \cos( \alpha )  +  \cos( \beta )  = 2  \cos \bigg(\dfrac{ \alpha  +  \beta }{2}  \bigg) \cos \bigg( \dfrac{ \alpha - \beta }{2}  \bigg)

 \rm \mapsto \sin( \alpha ) -  \sin( \beta )  = 2 \cos \bigg( \dfrac{ \alpha  +  \beta }{2}  \bigg)  \sin \bigg( \dfrac{ \alpha - \beta }{2} \bigg)

Taking LHS,

 \rm\dfrac{ \cos(7x) +  \cos(5x) }{ \sin(7x) -  \sin(5x)  }

 \rm =  \dfrac{2 \cos \bigg( \dfrac{7x + 5x}{2}\bigg) \cos \bigg( \dfrac{7x - 5x}{2} \bigg) }{2 \cos \bigg( \dfrac{7x + 5y}{2} \bigg) \sin \bigg( \dfrac{7x - 5x}{2} \bigg) }

 \rm =  \dfrac{ \cancel{2}  \cancel{\cos(6x)}\cos (x)}{ \cancel{2}  \cancel{\cos(6x)}\sin(x)}

 \rm =  \dfrac{ \cos(x) }{ \sin(x) }

 \rm =  \dfrac{1}{ \tan(x) }

 \rm =  \cot(x)

Taking RHS,

 \rm =  \cot(x)

Hence, LHS = RHS (Proved)

Relationship between different Trigo Functions:

  1. sin(x) = 1/cosec(x)
  2. cos(x) = 1/sec(x)
  3. tan(x) = 1/cot(x)
  4. sin(x)/cos(x) = tan(x)
Similar questions