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ale4655 [162]
3 years ago
10

Prove: cos(x+y)/cosxsiny=coty-tanx

Mathematics
2 answers:
Doss [256]3 years ago
8 0
Cos (x+y)/cos x sin y    

= cos x cos y - sin x sin y  / cos x sin y

= cos x cos y/cos x sin y     - sin x sin y / cos x sin y

hope this helps
Agata [3.3K]3 years ago
5 0

Answer:

Use Cosine Sum identity

Step-by-step explanation:

It's a simple demonstration if you know what trigonometric identity you should use. In this case use cosine sum identity which states:

cos(a+b)=cos(a)*cos(b)-sin(a)*sin(b)

Also keep in mind this property of fractions:

\frac{a \pm b}{c} =\frac{a}{c} \pm \frac{b}{c} \hspace{3} c\neq0

Using the previous information:

\frac{cos(x+y)}{cos(x)*sin(y)}=\frac{cos(x)*cos(y)-sin(x)*sin(y)}{cos(x)*sin(y)}=\frac{cos(x)*cos(y)}{cos(x)*sin(y)}-\frac{sin(x)sin(y)}{cos(x)*sin(y)}\\    \\=\frac{cos(y)}{sin(y)} -\frac{sin(x)}{cos(x)}

Also, according to the basic identities:

\frac{cos(\theta)}{sin(\theta)} =cot(\theta)\\\\\frac{sin(\theta)}{cos(\theta)}= tan(\theta)

Therefore:

\frac{cos(x+y)}{cos(x)*sin(y)}=\frac{cos(y)}{sin(y)} -\frac{sin(x)}{cos(x)}=cot(y)-tan(x)

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In order to find the common prime factors of 130 and 165, let us first find prime factors of 130 and 165 individually and then we can compare.

Prime factors of 130 can be found as:

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The proper factors of 130 are 1, 2, 5, 10, 13, 26, and 65.

The prime factors of 130 are 2, 5, and 13.

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165 is a composite number because it has factors other than 1 and itself. It is not a prime number.

The 8 factors of 165 are 1, 3, 5, 11, 15, 33, 55, and 165. The factor pairs of 165 are 1 x 165, 3 x 55, 5 x 33, and 11 x 15.

The proper factors of 165 are 1, 3, 5, 11, 15, 33, and 55.

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Answer: On comparing the two, we can see that 5 is the common prime factor of both 130 and 165.

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