Question

Cos9degrees+ Sin9degrees/Cos9degrees-Sin9degrees= Cot36degrees.

Answer 1

Answer:

LHS = (Cos9° + Sin9°) / (Cos9° - Sin9°)  

Divide by Nr and Dr  by  Cos9° ...

LHS = (1 + Tan9°) / (1 - Tan9°)

       = (Tan 45° + Tan 9°) / (1 - Tan 45° * Tan 9°)  

       = Tan (45°+9°)  

       = Tan 54°

       = Cot 36°.

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To prove : \frac{\cos 9+\sin 9}{\cos 9-\sin 9}=\cot 36

Proof :

Take LHS,

\frac{\cos 9+\sin 9}{\cos 9-\sin 9}

Take cos 9 common,

=\frac{1+\frac{\sin 9}{\cos 9}}{1-\frac{\sin 9}{\cos 9}}

=\frac{1+\tan 9}{1-\tan 9}

=\frac{\tan 45+\tan 9}{1-\tan 9\tan 45}

We know formula,

\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}

Here, A=45 and B=9

=\tan(45+9)

=\tan(54)

=\tan(90-36)

We know, \tan (90-\theta)=\cot \theta

=\cot 36

=RHS

Hence proved.

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