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Show that the values of sin2θcos2θsinθ+cosθ=0.

Answer
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Hint: In this question we have to show that the given trigonometric expression equals zero. For this we are going to show that by using trigonometric identities in angle and ratio and also we are going to multiply and add the trigonometric identities in complete step by step solution.
Trigonometric is a function that deals with the relationship between the sides and angles of triangles.

Formula used: There are six function of an angle commonly used in trigonometry, they are sine, cosine, tangent,  cosecant, secant, co - tangent. In this sum we are going to see about only sine and cosine angle and ratio formula. The formulas are
sin2θcos2θ=2sin(2θπ4)
sinθcosθ=2sin(θπ4)
sina + sinb=2sin(a+b2).cos(ab2)

Complete step-by-step answer:
Let consider the given equation as f(x)=sin2θcos2θsinθ+cosθ=0 .
Now, Rewrite the above equation as f(x)=(sin2θcos2θ)(sinθcosθ) = 0.
Here, we applying the trigonometric angles formulas on the expression, the sine and cosine angle are commonly known as sin and cos.
 sinθcosθ=2sin(θπ4)
Now, we get
sin2θcos2θ=2sin(2θπ4) and
sinθcosθ=2sin(θπ4)
Substitute the two trigonometric identities into the f(x), we get
f(x)=2sin(2θπ4)+2sin(θπ4)=0
Taking 2 as a common term in the above both terms, then
2(sin(2θπ4)+sin(θπ4))=0
Now, cross multiply the 2 into denominator of right hand side, we get
sin(2θπ4)+sin(θπ4)=02
We know that any number divisible by zero is zero. Then,
sin(2θπ4)+sin(θπ4)=0
Applying the trigonometry identity,
sina + sinb=2sin(a+b2).cos(ab2)
Now, consider here, a=2θπ4 and b=θπ4 . Then, we substitute in the above trigonometric identity, we get
sin(2θπ4)+sin(θπ4)=sin(2θπ4+θπ4).cos(2θπ4θ+π4)
After adding and subtracting the trigonometric identities, we get
sin(2θπ4)+sin(θπ4)=sin(3θ2π4).cos(θ)
sin(2θπ4)+sin(θπ4)=sin(3θπ2).cosθ
Since there are two solutions:
1. cosθ=0 if and only if θ=π2 and θ=3π2 .
2. sin(3θ2π2)=0
Now, we have to find the angle where the sine term becomes zero.
3θ2=π3θ=2π
θ=2π3 .
Also find another angle for sine term becomes zero.
3θ2=2π3θ=4π
θ=4π3 .
sin(3θ2π2)=0 if and only if θ=2π3 and θ=4π3.
Therefore, from the above angle which shows that the trigonometric identities become zero.
sin(2θπ4)+sin(θπ4)=0
Hence proved.

Note: All the trigonometric functions are positive in the first quadrant. SIn and Cosec are positive in the second quadrant. Tan and Cot are positive in the third quadrant. Cos and Sec are positive in the fourth quadrant.
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