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If 3tanθ=3sinθ, find the value of sin2θcos2θ.

Answer
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Hint: In this question, we have to find out the required trigonometric expression’s value from the given equation.
We need to first use the trigonometric formulas to bring the given equation in a shorter form so that we can find out the value of θ from the given equation then putting the value of θ in the given expression we will get the solution.
Trigonometric formula:
sin2θ+cos2θ=1
tanθ=sinθcosθ

Complete step-by-step solution:
The given trigonometric equation is 3tanθ=3sinθ.
We need to find out the value of sin2θcos2θ.
Now, we have to first find out the value of θ from the given equation.
We have,
3tanθ=3sinθ
We know, tanθ=sinθcosθ.
Putting the formula in the given equation we get,
3sinθcosθ=3sinθ
By cross multiplication we get,
sinθcosθ×1sinθ=33
Solving the equation we get,
1cosθ=3
cosθ=13
Squaring we get,
cos2θ=13
We know,sin2θ+cos2θ=1
sin2θ=1cos2θ
sin2θ=113
sin2θ=313=23
Hence, sin2θcos2θ=2313
sin2θcos2θ=13

Hence, the value of sin2θcos2θ is 13.

Note: Sin Cos formulas are based on sides of the right-angled triangle. Sin and Cos are basic trigonometric functions along with tan function, in trigonometry. Sine of angle is equal to the ratio of opposite side and hypotenuse whereas cosine of an angle is equal to ratio of adjacent side and hypotenuse.
sinθ=Opposite sideHypotenuse
cosθ=AdjacentHypotenuse
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In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. The most widely used trigonometric functions are the sine, the cosine, and the tangent.