Solving Trigonometric Equations:
\[ \csc^2x - 2\cot^2x = 0 \]
Answer (2)
cs c 2 x − 2 co t 2 x = 0 ; D : x = kπ ( k ∈ Z ) s i n 2 x 1 − 2 ⋅ s i n 2 x co s 2 x = 0 s i n 2 x 1 − 2 co s 2 x = 0 ⟺ 1 − 2 co s 2 x = 0 − 2 co s 2 x = − 1 / : ( − 2 ) co s 2 x = 2 1 cos x = 2 1 ∨ cos x = − 2 1 cos x = 2 2 ∨ cos x = − 2 2
x = 4 π + 2 kπ ∨ x = − 4 π + 2 kπ ∨ x = 4 3 π + 2 kπ ∨ x = − 4 3 π + 2 kπ A n s w er : x = 4 π + 2 kπ ( k ∈ Z )
To solve csc 2 x − 2 cot 2 x = 0 , we substitute the definitions of cosecant and cotangent, leading to the equation cos 2 x = 2 1 . This gives solutions of x = 4 π + k 2 π for integer values of k .
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