Question

To prove :One plus cot square theta into tan theta by sec square theta = cot theta

Answer 1

we are given

`((1+cot^2(\theta))*tan(\theta))/(sec^2(\theta)) =cot(\theta)`

We will simplify left side and make it equal to right side

Left side:

`((1+cot^2(\theta))*tan(\theta))/(sec^2(\theta))`

we can use trigonometric identity

`1+cot^2(\theta)=csc^2(\theta)`

we can replace it

`((csc^2(\theta))*tan(\theta))/(sec^2(\theta))`

we know that

csc=1/sin and sec=1/cos

so, we can replace it

and we get

`(cos^2(\theta)tan(\theta))/(sin^2(\theta))`

now, we know that

tan =sin/cos

`(cos^2(\theta)*sin(\theta))/(sin^2(\theta)*cos(\theta))`

we can simplify it

and we get

`(cos(\theta))/(sin(\theta))`

we can also write it as

`=cot(\theta)`

Right Side:

`cot(\theta)`

we can see that

left side = right side

so,

`((1+cot^2(\theta))*tan(\theta))/(sec^2(\theta)) =cot(\theta)`......Answer