Question

cos11 + sin11 cos11 - sin11 + = tan 56

Answer 1

Step-by-step explanation:

cos11−sin11cos11+sin11=tan56 or cot34

Given:

\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34cos11−sin11cos11+sin11=tan56 or cot34

To Prove:

\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34cos11−sin11cos11+sin11=tan56 or cot34

Solution:

RHS

\frac{\cos 11+\sin 11}{\cos 11-\sin 11}cos11−sin11cos11+sin11

Divide numerator and denominator by cos⁡11

Hence the equation becomes

\frac{1+\tan 11}{1-\tan 11} \ldots(1)1−tan111+tan11…(1)

By formula,

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

Bringing eq(1) in the above form,

\frac{1+\tan 11}{1-\tan 11}=\frac{\tan 45+\tan 11}{1-(\tan 45)(\tan 11)}1−tan111+tan11=1−(tan45)(tan11)tan45+tan11

\frac{\tan 45+\tan 11}{1-(\tan 45)(\tan 11)}=\tan (45+11)=\tan (56)1−(tan45)(tan11)tan45+tan11=tan(45+11)=tan(56)

\begin{gathered}\begin{array}{l}{\tan 56=\tan (90-34)} \\ {\tan (90-\mathrm{A})=\cot \mathrm{A}}\end{array}\end{gathered}tan56=tan(90−34)tan(90−A)=cotA

Hence \tan (90-34)=\cot 34tan(90−34)=cot34 = RHS  

\bold{\frac{\cos 11+\sin 11}{\cos 11-\sin 11}=\tan 56 \text { or } \cot 34}cos11−sin11cos11+sin11=tan56 or cot34

LHS = RHS  

Hence proved.