Question

4×sin2a=(1-cos4a)×(ctga+tga)​

Answer 1

$4sin2a=(1-(1-2sin^22a))*(\frac{cos^2a+sin^2a}{sina*cosa} );\\\left \{ {{4sin2a=\frac{2sin^22a*1}{\frac{sin2a}{2} }} \atop {{\left[\begin{array}{ccc}sina\neq 0\\cosa\neq 0\\\end{array} }}\right.$

$\left \{ {{4sin2a=\frac{4sin^22a}{sin2a} =4sin2a} \atop {{\left[\begin{array}{ccc}sina\neq 0\\cosa\neq 0\\sin2a\neq 0\end{array}}} \right.$

${{\left[\begin{array}{ccc}sina\neq 0\\cosa\neq 0\\2*cosa*sina\neq 0\end{array}}}$

Ответ: x≠{pi*n/2}, n∈Z.

Answer 2

To solve the trigonometric equation 4×sin^2(a) = (1-cos^4(a))×(cot(a)+tan(a)), we'll first manipulate the right-hand side of the equation using trigonometric identities and then solve for the variable "a."

Step 1: Trigonometric identities Recall the following trigonometric identities:

• sin^2(a) = 1 - cos^2(a)
• cot(a) = 1/tan(a)

Step 2: Substitute the identities Replace sin^2(a) and cot(a) with the corresponding identities: 4×(1 - cos^2(a)) = (1 - cos^4(a)) × (1/tan(a) + tan(a))

Step 3: Convert tan(a) to sin(a)/cos(a) Replace tan(a) with sin(a)/cos(a): 4×(1 - cos^2(a)) = (1 - cos^4(a)) × (1/(sin(a)/cos(a)) + sin(a)/cos(a))

Step 4: Simplify the right-hand side To add fractions, find a common denominator, which is cos(a): 4×(1 - cos^2(a)) = (1 - cos^4(a)) × ((cos(a))/(sin(a)) + (sin(a)×cos(a))/(cos(a)))

Step 5: More simplification Now, simplify further: 4×(1 - cos^2(a)) = (1 - cos^4(a)) × ((cos(a) + sin^2(a))/(sin(a)))

Step 6: Replace 1 - cos^2(a) with sin^2(a) Use the identity 1 - cos^2(a) = sin^2(a): 4×sin^2(a) = (1 - cos^4(a)) × ((cos(a) + sin^2(a))/(sin(a)))

Step 7: Cancel out sin(a) from the right-hand side Cancel out sin(a) from the denominator and the sin^2(a) term in the numerator: 4×sin^2(a) = (1 - cos^4(a)) × (cos(a) + sin(a))

Step 8: Expand the right-hand side Use the distributive property to expand the right-hand side: 4×sin^2(a) = cos(a) - cos^5(a) + sin(a) - sin(a)×cos^4(a)

Step 9: Move all terms to one side of the equation Now, move all terms to one side of the equation: 4×sin^2(a) - cos(a) + cos^5(a) - sin(a) + sin(a)×cos^4(a) = 0

Step 10: Combine like terms Combine the terms with sin(a) and simplify: 4×sin^2(a) - cos(a) + cos^5(a) + sin(a)×(1 - cos^4(a)) = 0

Step 11: Use the identity sin^2(a) + cos^2(a) = 1 Replace sin^2(a) in terms of cos^2(a) using the identity sin^2(a) = 1 - cos^2(a): 4×(1 - cos^2(a)) - cos(a) + cos^5(a) + sin(a)×(1 - cos^4(a)) = 0

Step 12: Replace sin(a) with √(1 - cos^2(a)) Use the identity sin(a) = √(1 - cos^2(a)): 4×(1 - cos^2(a)) - cos(a) + cos^5(a) + √(1 - cos^2(a))×(1 - cos^4(a)) = 0

At this point, we have a trigonometric equation involving cos(a) and its powers. To find the values of "a" that satisfy this equation, we can use numerical methods or a graphing calculator. Unfortunately, the equation is quite complex, and an exact analytical solution may not be feasible.