If sin2A=2sinA.cosA and sin20°=k ,then the value of cos20°.cos20°cos80°.cos160°=?

To find the value of cos20° · cos20° · cos80° · cos160°, we can start by simplifying the expression using trigonometric identities. Given that sin²A = 2sinA · cosA, we can recognize that this is a double angle identity for sine, specifically sin²A = sin(2A). So, let’s break this down step by step to find the result.

Understanding the Trigonometric Identities

First, let’s rewrite the equation you provided. We know that:

• sin²A = sin(2A) implies that sin²20° = sin(40°).

Given that sin20° = k, we can express sin²20° as k². Thus, we have:

• sin(40°) = k².

Cosine Values

Now, to find the value of cos20° · cos20° · cos80° · cos160°, we can use the following relationships:

• cos(80°) = sin(10°) because of the co-function identity: cos(θ) = sin(90° - θ).
• cos(160°) = -cos(20°) as it is in the second quadrant.

Substituting Values

This allows us to rewrite our original expression:

• cos20° · cos20° · cos80° · cos160° = cos²20° · sin(10°) · (-cos(20°)).

Thus, we can simplify this to:

• = -cos³20° · sin(10°).

Finding sin(10°)

To find the value of sin(10°), we can use the identity sin(2A) = 2sin(A)cos(A). Here, we can express sin(20°) using 10°:

• sin(20°) = 2sin(10°)cos(10°).

Knowing sin(20°) = k, we have:

• k = 2sin(10°)cos(10°).

This implies that:

• sin(10°) = k/(2cos(10°)).

Final Expression

Now, substituting sin(10°) back into our expression:

• -cos³20° · (k/(2cos(10°))).

So our final equation becomes:

• = -k/(2cos(10°)) · cos³20°.

Conclusion of the Calculation

To summarize, we have simplified the expression step-by-step, utilizing trigonometric identities to arrive at:

• cos20° · cos20° · cos80° · cos160° = -k/(2cos(10°)) · cos³20°.

To find an exact numerical value, we would need the specific value for k and cos(10°), but this gives you a clear path to evaluate your expression based on the information provided. If you have specific numeric values for k or wish to explore further, feel free to ask!