Prove the trigonometric identity:
\[ \sec^2x + \csc^2x = (\sec^2 x)(\csc^2 x) \]
Answer (2)
se c 2 x + cs c 2 x = ( se c 2 x ) ( cs c 2 x ) L = co s 2 x 1 + s i n 2 x 1 = co s 2 x s i n 2 x s i n 2 x + co s 2 x s i n 2 x co s 2 x = co s 2 x s i n 2 x s i n 2 x + co s 2 x = co s 2 x s i n 2 x 1 = co s 2 x 1 ⋅ s i n 2 x 1 = se c 2 x ⋅ cs c 2 x = R
We proved the identity sec 2 x + csc 2 x = ( sec 2 x ) ( csc 2 x ) by expressing secant and cosecant in terms of sine and cosine. After rewriting the left side, applying a common denominator, and using the Pythagorean identity, we simplified to the right side of the equation. Hence, the identity holds true.
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