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

we are given

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

Left side:

we can use trigonometric identity

we can replace it

we know that

csc=1/sin and sec=1/cos

so, we can replace it

and we get

now, we know that

tan =sin/cos

we can simplify it

and we get

we can also write it as

Right Side:

we can see that

left side = right side

so,

......Answer

sec ф = 5/3 (quadrant 4)

let x = base

y = perp

z = hypotenuse

sec ф = hypotenuse/base = 5/3

⇒ z = 5 and x = 3

finding y:

z² = x² + y²

5² = 3² + y²

y² = 25-9

y² = 16

y = 4

now finding all other geometric functions of theta:

sin ф = y/z = 4/5

sin ф = -4/5 (sin is negative in the 4th quadrant)

cos ф = x/z = 3/5

cos ф = 3/5 (cos is positive in 4th quadrant)

tan ф = y/x = 4/3

tan ф = -4/3 (tan is negative in 4th quadrant)

cosec ф = z/y = 5/4

cosec ф = -5/4 (cosec is negative in the 4th quadrant)

cot ф = x/y = 3/4

tan ф = -3/4 (tan is negative in 4th quadrant)

it is 10.6 according to my calculations