Question

Prove the identity ((secy-tany) (secy+ tan y))/secy= COS U

Answer 1

boring

Explanation:

To prove the given identity, we'll start with the left side and simplify it step by step until we obtain the right side of the equation:

We have:

((sec y - tan y)(sec y + tan y)) / sec y

Using the difference of squares, we can rewrite the numerator:

((sec y)^2 - (tan y)^2) / sec y

Recall the trigonometric identity: (sec x)^2 - (tan x)^2 = 1

Substituting this identity into our expression:

(1) / sec y

Using the reciprocal identity: sec x = 1 / cos x

We can rewrite the expression as:

1 / (1 / cos y)

Simplifying further by multiplying by the reciprocal:

cos y / 1

Which simplifies to:

cos y

Therefore, we have shown that the left side of the identity ((sec y - tan y)(sec y + tan y)) / sec y simplifies to cos y, which is the same as the right side of the equation.