Question

evaluate 8/cot^2-8/cos^2

Answer 1

The value of $\dfrac{8}{\cot^2 A}-\dfrac{8}{\cos^2 A}$ $=8\csc^2 A}$.

Step-by-step explanation:

We have,

$\dfrac{8}{\cot^2 A}-\dfrac{8}{\cos^2 A}$

To find, the value of $\dfrac{8}{\cot^2 A}-\dfrac{8}{\cos^2 A}$ = ?

∴ $\dfrac{8}{\cot^2 A}-\dfrac{8}{\cos^2 A}$

Using the trigonometric identity,

$\cot A=\dfrac{\cos A}{\sin A}$

$=\dfrac{8}{\dfrac{\cos^2 A}{\sin^2 A} }-\dfrac{8}{\cos^2 A}$

$=\dfrac{8}{\cos^2 A \sin^2 A}-\dfrac{8}{\cos^2 A}$

Taking commonas $\dfrac{8}{\cos^2 A}$, we get

=$\dfrac{8}{\cos^2 A}(\dfrac{1}{ \sin^2 A}-1)$

$=\dfrac{8}{\cos^2 A}(\dfrac{1-\sin^2 A}{\sin^2 A})$

Using the trigonometric identity,

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

$=\dfrac{8}{\cos^2 A}(\dfrac{\cos^2 A}{\sin^2 A})$

$=\dfrac{8}{\sin^2 A}$

$=8\csc^2 A}$

Thus, the value of $\dfrac{8}{\cot^2 A}-\dfrac{8}{\cos^2 A}$ $=8\csc^2 A}$.

Answer 2

Cot^2 can be written as cos^2/sin^2

And cos^2 can be written as 1/sec^2

So now the denominator changes

8/cos^2/sin^2-8/1/sec^2

Now , the step follows

8(sin^2)/cos^2-8(sec^2)/1

8(tan^2)-8(sec^2)

Taking 8 as common

8(tan^2-sec^2)

We know that sec^2-tan^2=1

Tan^2-sec^2=-1

8(-1)=-8

Therefore, 8/cot^2-8/cos^2=-8