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DochEvi [55]
3 years ago
7

Which of the following is equivalent to tan2θcos(2θ) for all values of θ for which tan2θcos(2θ) is defined?

Mathematics
1 answer:
Aloiza [94]3 years ago
7 0

Answer:

2sin²θ - tan²θ

Step-by-step explanation:

Given

tan²θcos(2θ)

Required

Simplify

We start by simplifying cos(2θ)

cos(2θ) = cos(θ+θ)

From Cosine formula

cos(A+A) = cosAcosA - sinAsinA

cos(A+A) = cos²A - sin²A

By comparison

cos(2θ) = cos(θ+θ)

cos(2θ) = cos²θ - sin²θ ----- equation 1

Recall that cos²θ + sin²θ = 1

Make sin²θ the subject of formula

sin²θ = 1 - cos²θ

Substitute sin²θ = 1 - cos²θ in equation 1

cos(2θ) = cos²θ - (1 - cos²θ)

cos(2θ) = cos²θ - 1 +cos²θ

cos(2θ) = cos²θ + cos²θ - 1

cos(2θ) = 2cos²θ - 1

Substitute 2cos²θ - 1 for cos(2θ) in the given question

tan²θcos(2θ) becomes

tan²θ(2cos²θ - 1)

Open brackets

2cos²θtan²θ - tan²θ

------------------------

Simplify tan²θ

tan²θ = (tanθ)²

Recall that tanθ =  sinθ/cosθ

So, we have

tan²θ = (sinθ/cosθ)²

tan²θ = sin²θ/cos²θ

------------------------

Substitute sin²θ/cos²θ for tan²θ

2cos²θtan²θ - tan²θ becomes

2cos²θ(sin²θ/cos²θ) - tan²θ

Open bracket (cos²θ will cancel out cos²θ) to give

2(sin²θ) - tan²θ

2sin²θ - tan²θ

Hence, the simplification of tan²θcos(2θ) is 2sin²θ - tan²θ

Option E is correct

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Step-by-step explanation: For this question you can find the answer by taking 52.50 ÷ 21 which is 2.5 so that packs are $2.50 a pack and then you can either plug in numbers or take 17.50 ÷ 2.5.

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Softa [21]

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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let me divide your question into two parts A and B as;

A.Rewrite the expression using the GCF and distributive property. (110-44)

B.how many 24-foot jump ropes can be made from a rope that is 100 feet long?

Answer:

A.(110-44)=22(5-2)

B. 4 jump ropes of 24 feet length

Step-by-step explanation:

A.Rewrite the expression using the GCF and distributive property. (110-44

Answer:

Find the GCF of 110 and 44 then take that GCF as common factor out of (110-44) then we can get new expression for (110-44) as 22(5-2) i.e.

prime factorization of 110=2x5x11

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GCf is the product of factors that appear in both of the prime factorization which is 2x11=22

thus using GCF (22) as common factor we can rewrite the given expression as

22(5-2) which is distributive law of multiplication over subtraction.

B.how many 24-foot jump ropes can be made from a rope that is 100 feet long?

Answer:

Given

Total length of the rope =100 feet

length of a jump rope =24

solution:

For one jump-rope we need 24 feet

For two jump-ropes we need 48 feet

For three jump-ropes we need 72 feet

For four jump-ropes we need 96 feet

thus from a 100 feet long rope we can make 4 jump-ropes of 24 feet length with 6 feet remaining as extra.

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Answer:

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Step-by-step explanation:

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