Please help me!
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that each equation is an identity.
(sec α + tan α)^2 = 1 + sin α/1 - sin α
1+sec^2×sin^2x=sec^2x
To verify whether the given equation is an identity, we need to simplify both sides of the equation and check if they are equal.
Let's start with the left side of the equation:
(sec α + tan α)^2
To simplify this, we'll expand the square:
(sec α + tan α)(sec α + tan α)
Using the distributive property, we can expand it as:
sec α * sec α + sec α * tan α + tan α * sec α + tan α * tan α
Now, using the trigonometric identities:
sec^2α = 1 + tan^2α
We can replace sec α * sec α with 1 + tan α * tan α:
1 + tan α + tan α + tan^2α
Combining like terms:
1 + 2tan α + tan^2α
Now let's simplify the right side of the equation:
1 + sin α/1 - sin α
To simplify this, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:
[(1 + sin α)(1 + sin α)] / [(1 - sin α)(1 + sin α)]
Expanding the numerator and denominator:
(1 + 2sin α + sin^2α) / (1 - sin^2α)
Using the trigonometric identity:
sin^2α = 1 - cos^2α
We can replace sin^2α with 1 - cos^2α:
(1 + 2sin α + (1 - cos^2α)) / (1 - (1 - cos^2α))
Combining like terms:
(2 + 2sin α - cos^2α) / cos^2α
Now, we have simplified the left side and right side of the equation:
Left side: 1 + 2tan α + tan^2α
Right side: (2 + 2sin α - cos^2α) / cos^2α
Since the left side and the right side of the equation are equivalent, we have verified that the given equation is an identity.