Question

prove that cos11+sin11/cos11-sin11=tan56 or cot 34

Answer 1

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Answer 2

$\bold{\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34}$

Given:

$\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34$

To Prove:

$\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34$

Solution:

RHS

$\frac{\cos 11+\sin 11}{\cos 11-\sin 11}$

Divide numerator and denominator by cos⁡11

Hence the equation becomes

$\frac{1+\tan 11}{1-\tan 11} \ldots(1)$

By formula,

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

Bringing eq(1) in the above form,

$\frac{1+\tan 11}{1-\tan 11}=\frac{\tan 45+\tan 11}{1-(\tan 45)(\tan 11)}$

$\frac{\tan 45+\tan 11}{1-(\tan 45)(\tan 11)}=\tan (45+11)=\tan (56)$

$\begin{array}{l}{\tan 56=\tan (90-34)} \\ {\tan (90-\mathrm{A})=\cot \mathrm{A}}\end{array}$

Hence $\tan (90-34)=\cot 34$= RHS  

$\bold{\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34}$

LHS = RHS  

Hence proved.