Question

सिद्ध कीजिये कि (cos7x+cos5)/(sin7x-sin5)=cot x​

Answer 1

$\underline{\textsf{To prove:}}$

$\mathsf{\dfrac{cos7x+cos5x}{sin7x-sin5x}=cotx}$

$\underline{\textsf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{cos7x+cos5x}{sin7x-sin5x}}$

$\mathsf{Using\;the\;following\;identities}$

$\boxed{\mathsf{cosC+cosD=2\,cos\left(\dfrac{C+D}{2}\right)\,cos\left(\dfrac{C-D}{2}\right)}}$

$\boxed{\mathsf{sinC-sinD=2\,cos\left(\dfrac{C+D}{2}\right)\,sin\left(\dfrac{C-D}{2}\right)}}$

$\mathsf{=\dfrac{2\,cos\left(\dfrac{7x+5x}{2}\right)\,cos\left(\dfrac{7x-5x}{2}\right)}{2\,cos\left(\dfrac{7x+5x}{2}\right)\,sin\left(\dfrac{7x-5x}{2}\right)}}$

$\mathsf{=\dfrac{2\,cos\left(\dfrac{12x}{2}\right)\,cos\left(\dfrac{2x}{2}\right)}{2\,cos\left(\dfrac{12x}{2}\right)\,sin\left(\dfrac{2x}{2}\right)}}$

$\mathsf{=\dfrac{2\,cos6x\,cosx}{2\,cos6x\,sinx}}$

$\mathsf{=\dfrac{cosx}{sinx}}$

$\mathsf{=cotx}$

$\implies\boxed{\mathsf{\dfrac{cos7x+cos5x}{sin7x-sin5x}=cotx}}$

$\underline{\textsf{Find more:}}$

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