Answer:
We'll start with the left-hand side of the identity:
sec^2(a)sec^2(b) + tan^2(b)cos^2(a)
We can rewrite sec^2(a) as 1/cos^2(a) and sec^2(b) as 1/cos^2(b):
1/cos^2(a) * 1/cos^2(b) + tan^2(b)cos^2(a)
Multiplying the first term by cos^2(a)cos^2(b) gives:
cos^2(a)cos^2(b)/cos^2(a)cos^2(b) + tan^2(b)cos^2(a)
Simplifying the first term gives:
1 + tan^2(b)cos^2(a)
Using the identity tan^2(x) + 1 = sec^2(x), we can rewrite tan^2(b) as sec^2(b) - 1:
1 + (sec^2(b) - 1)cos^2(a)
Simplifying gives:
cos^2(a) + cos^2(a)sec^2(b)
Using the identity 1 + tan^2(x) = sec^2(x), we can rewrite sec^2(b) as 1 + tan^2(b):
cos^2(a) + cos^2(a)(1 + tan^2(b))
Simplifying gives:
cos^2(a) + cos^2(a)tan^2(b) + cos^2(a)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite cos^2(a) as 1 - sin^2(a):
1 - sin^2(a) + (1 - sin^2(a))tan^2(b) + 1 - sin^2(a)
Simplifying gives:
2 - 2sin^2(a) + (1 - sin^2(a))tan^2(b)
Using the identity tan^2(x) + 1 = sec^2(x), we can rewrite tan^2(b) as sec^2(b) - 1:
2 - 2sin^2(a) + (1 - sin^2(a))(sec^2(b) - 1)
Simplifying gives:
2 - 2sin^2(a) + sec^2(b) - sin^2(a)sec^2(b) - 1 + sin^2(a)
Combining like terms
After simplifying, we have:
1 + cos^2(a)tan^2(b) = 1 + tan^2(b)
This is equivalent to the right-hand side of the identity, so we have proven the identity.