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Question

# Assertion :Numerical value of expression x=4log52+log5sinπ5+log5sin2π5+log5sin3π5+log5sin4π5 is 1. Reason: sin18o=14(√5−1) and cos36o=14(√5+1).

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
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C
Assertion is correct but Reason is incorrect.
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D
Assertion is incorrect but Reason is correct.
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.We have, x=4log52+log5sinπ5+log5sin2π5+log5sin3π5+log5sin4π5Using sin3π5=sin2π5;sin4π5=sinπ5 we getx=4log52+log5(sin2π5sin22π5)[∵loga+logb+logc+logd=log(abcd)]x=4log52+log5[(1−cos236o)(1−cos272o)]x=4log52+log5(1−cos236o)(1−sin218o)x=4log52+log5[1−116(√5+1)2][1−116(√5−1)2]x=4log52+log5[(10−2√5)(10+2√5)256]x=4log52+log5(80256)x=4log52+log5(524)⇒x=4log52+log55−4log52⇒x=1Hence, assertion is correct.Also, we know thatsin18o=14(√5−1) and cos36o=14(√5+1)Hence, reason is the correct statement and is the correct explanation for assertion.

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