Prove that tan^2a + cot^2a = 1 is not a trigonometric identity by producing a counterexample

Proof is given below.

Step-by-step explanation:

To prove: is not a trigonometric identity.

A trigonometric identity is an equation whose left hand side is always to the right hand side for any value of the given angle.

A counterexample is a method used to counter the given statement by taking a random value for the given quantity and disproving the left and right side of the equation.

So, let us take A = 45°

Then, left hand side of the equation becomes;

Therefore, the value of left hand side of the equation on plugging in 45° for A gives the result as 2.

But the right hand side of the equation is equal to 1.

Therefore,

So, this violates the given equation and hence the given equation is not always true. So, it's not a trigonometric identity.

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is