To show that the left-hand side (LHS) of the equation is equivalent to the right-hand side (RHS) by applying trigonometric identities, we'll start with the LHS and simplify it step by step.
LHS: 1 + sec^2(x) ⋅ sin^2(x)
Step 1: Rewrite sec^2(x) as 1 + tan^2(x) using the identity sec^2(x) = 1 + tan^2(x).
LHS: 1 + (1 + tan^2(x)) ⋅ sin^2(x)
Step 2: Distribute sin^2(x) into the parentheses.
LHS: 1 + sin^2(x) + tan^2(x) ⋅ sin^2(x)
Step 3: Rewrite tan^2(x) as sin^2(x)/cos^2(x) using the identity tan^2(x) = sin^2(x)/cos^2(x).
LHS: 1 + sin^2(x) + (sin^2(x)/cos^2(x)) ⋅ sin^2(x)
Step 4: Simplify the expression by multiplying the fractions.
LHS: 1 + sin^2(x) + (sin^4(x))/cos^2(x)
Step 5: Combine the terms with sin^2(x).
LHS: 1 + (1 + sin^4(x))/cos^2(x)
Step 6: Rewrite 1 as cos^2(x)/cos^2(x) to have a common denominator.
LHS: cos^2(x)/cos^2(x) + (1 + sin^4(x))/cos^2(x)
Step 7: Combine the fractions.
LHS: (cos^2(x) + 1 + sin^4(x))/cos^2(x)
Step 8: Rewrite cos^2(x) as 1 - sin^2(x) using the identity cos^2(x) = 1 - sin^2(x).
LHS: (1 - sin^2(x) + 1 + sin^4(x))/cos^2(x)
Step 9: Combine the terms.
LHS: (2 - sin^2(x) + sin^4(x))/cos^2(x)
Step 10: Rewrite sin^4(x) as (sin^2(x))^2.
LHS: (2 - sin^2(x) + (sin^2(x))^2)/cos^2(x)
Step 11: Factor the numerator.
LHS: (2 - sin^2(x) + sin^2(x))(1 + sin^2(x))/cos^2(x)
Step 12: Simplify.
LHS: (2)(1 + sin^2(x))/cos^2(x)
Step 13: Cancel out the common factor of 2.
LHS: (1 + sin^2(x))/cos^2(x)
Step 14: Rewrite 1 + sin^2(x) as cos^2(x) using the identity 1 + sin^2(x) = cos^2(x).
LHS: cos^2(x)/cos^2(x)
Step 15: Simplify.
LHS: 1
Since the left-hand side (LHS) simplifies to 1 and the right-hand side (RHS) is equal to 1, we have shown that LHS = RHS.
