Prove tan^2 theta - tan^2 phi = (sin^2 theta- sin^2 phi) /cos^2 theta cos^2phi

tan^2 theta - tan^2 phi = (sin^2 theta- sin^2 phi) /(cos^2 theta cos^2phi) (identity has been verified)

Step-by-step explanation:

Verify the following identity:

tan(θ)^2 - tan(ϕ)^2 = (sin(θ)^2 - sin(ϕ)^2)/(cos(θ)^2 cos(ϕ)^2)

Hint: | Eliminate the denominator on the right hand side.

Multiply both sides by cos(θ)^2 cos(ϕ)^2:

cos(θ)^2 cos(ϕ)^2 (tan(θ)^2 - tan(ϕ)^2) = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Express the left hand side in terms of sine and cosine.

Write tangent as sine/cosine:

cos(θ)^2 cos(ϕ)^2 ((sin(θ)/cos(θ))^2 - (sin(ϕ)/cos(ϕ))^2) = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Simplify the left hand side.

cos(θ)^2 cos(ϕ)^2 ((sin(θ)/cos(θ))^2 - (sin(ϕ)/cos(ϕ))^2) = cos(θ)^2 cos(ϕ)^2 ((sin(θ)^2)/(cos(θ)^2) - (sin(ϕ)^2)/(cos(ϕ)^2)):

cos(θ)^2 cos(ϕ)^2 (sin(θ)^2/cos(θ)^2 - sin(ϕ)^2/cos(ϕ)^2) = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Put the fractions in sin(θ)^2/cos(θ)^2 - sin(ϕ)^2/cos(ϕ)^2 over a common denominator.

Put sin(θ)^2/cos(θ)^2 - sin(ϕ)^2/cos(ϕ)^2 over the common denominator cos(θ)^2 cos(ϕ)^2: sin(θ)^2/cos(θ)^2 - sin(ϕ)^2/cos(ϕ)^2 = (cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2)/(cos(θ)^2 cos(ϕ)^2):

cos(θ)^2 cos(ϕ)^2 (cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2)/(cos(θ)^2 cos(ϕ)^2) = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Cancel down (cos(θ)^2 cos(ϕ)^2 (cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2))/(cos(θ)^2 cos(ϕ)^2).

Cancel cos(θ)^2 cos(ϕ)^2 from the numerator and denominator. (cos(θ)^2 cos(ϕ)^2 (cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2))/(cos(θ)^2 cos(ϕ)^2) = (cos(θ)^2 cos(ϕ)^2 (cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2))/(cos(θ)^2 cos(ϕ)^2) = cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2:

cos(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2 = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Express cos(ϕ)^2 in terms of sine via the Pythagorean identity.

cos(ϕ)^2 = 1 - sin(ϕ)^2:

1 - sin(ϕ)^2 sin(θ)^2 - cos(θ)^2 sin(ϕ)^2 = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Express cos(θ)^2 in terms of sine via the Pythagorean identity.

cos(θ)^2 = 1 - sin(θ)^2:

sin(θ)^2 (1 - sin(ϕ)^2) - 1 - sin(θ)^2 sin(ϕ)^2 = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Expand (1 - sin(ϕ)^2) sin(θ)^2.

(1 - sin(ϕ)^2) sin(θ)^2 = sin(θ)^2 - sin(θ)^2 sin(ϕ)^2:

sin(θ)^2 - sin(θ)^2 sin(ϕ)^2 - sin(ϕ)^2 (1 - sin(θ)^2) = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Expand -(1 - sin(θ)^2) sin(ϕ)^2.

-(1 - sin(θ)^2) sin(ϕ)^2 = sin(θ)^2 sin(ϕ)^2 - sin(ϕ)^2:

sin(θ)^2 - sin(θ)^2 sin(ϕ)^2 + sin(θ)^2 sin(ϕ)^2 - sin(ϕ)^2 = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Evaluate sin(θ)^2 - sin(θ)^2 sin(ϕ)^2 - sin(ϕ)^2 + sin(θ)^2 sin(ϕ)^2.

sin(θ)^2 - sin(θ)^2 sin(ϕ)^2 - sin(ϕ)^2 + sin(θ)^2 sin(ϕ)^2 = sin(θ)^2 - sin(ϕ)^2:

sin(θ)^2 - sin(ϕ)^2 = ^?sin(θ)^2 - sin(ϕ)^2

Hint: | Come to a conclusion.

The left hand side and right hand side are identical:

(identity has been verified)

the population of other specie of birds will decline or other specie of birds will find some alternate source of food to survive.

explanation:

according to the competitive exclusion principle, which is also known as gause's law, two specie living in the same niche and feeding on the same resource cannot co-exist stably.

as we have well studied the theory of natural selection which states that only those organisms survive which have better tendency to compete for food and adapt to the environmental conditions.

coming towards the question, what wi ll happen when one species outcompetes the other for the small fish , the other specie will not get enough food and will start to die or decline in population. eventually either other specie needs to find another alternative food source for itself or migrate at some other location where there is less competition otherwise it will just not exist anymore at that niche.

hope it !