Question

evaluate cot5 cot10 cot15 cot60 cot75 cot80 cot85/(cos^2 20+cos^70)+2

Answer 1

Answer:

root3/2

Step-by-step explanation:

all the cot terms will cancel after it will change to tan but cot 60 will be remaining =root3

cos20^2 +cos70^2 will become

cos20^2 +sin20^2 =1

Answer 2

The value of $\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot 75 \cot 80 \cot 85}{\cos^2 {20}+\cos^2{70}} +2$ = $\dfrac{1}{\sqrt{3}} +2$

Step-by-step explanation:

We have,

$\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot 75 \cot 80 \cot 85}{\cos^2 {20}+\cos^2{70}} +2$

To find, the value of $\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot 75 \cot 80 \cot 85}{\cos^2 {20}+\cos^2{70}} +2$ = ?

∴ $\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot 75 \cot 80 \cot 85}{\cos^2 {20}+\cos^2{70}} +2$

$=\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot (90-15) \cot (90-10) \cot (90-5)}{\cos^2 {(90-70)}+\cos^2{70}} +2$

$=\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \tan 15 \tan 10 \tan 5}{\sin^2 {70}+\cos^2{70}} +2$

Using the trigonometric identity,

$\tan A=\cot (90-A)$ and

$\tan A=\cot (90-A)$

$=\dfrac{(\cot 5.\tan 5)(\cot 10\tan 10) (\tan 15\cot 15)\cot 60}{1} +2$

Using the trigonometric identity,

$\sin^2 A+\cos^2A=1$

Using the trigonometric identity,

$\cot A.\tan A=1$

$=\cot 60 +2$

= $\dfrac{1}{\sqrt{3}} +2$

Thus, the value of $\dfrac{\cot 5 \cot 10 \cot 15 \cot 60 \cot 75 \cot 80 \cot 85}{\cos^2 {20}+\cos^2{70}} +2$ = $\dfrac{1}{\sqrt{3}} +2$