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katovenus [111]
2 years ago
7

Verify the identity cot(x-pi/2)=-tan x

Mathematics
1 answer:
vampirchik [111]2 years ago
4 0
To verify the identity we need the following identities:

i) \displaystyle{ \cot(x)= \frac{\cos x}{\sin x}

ii) \displaystyle{ \sin (x-y)=\ sinx\cdot\ cosy -\ siny\cdot\ cosx

iii) \displaystyle{ \cos (x-y)=\ cosx\cdot \ cosy +\ sinx\cdot\ siny.

Also, we have know that \displaystyle{ \sin \frac{ \pi }{2}=1 and \displaystyle{ \cos \frac{ \pi }{2}=0.


Thus, \displaystyle{ \cot(x-\frac{\pi}{2})= \frac{\cos (x-\frac{\pi}{2})}{\sin (x-\frac{\pi}{2})}

By (ii) and (iii) we have:

\displaystyle{ \frac{\cos (x-\frac{\pi}{2})}{\sin (x-\frac{\pi}{2})}= \frac{\ cosx\cdot \ cos\frac{\pi}{2} +\ sinx\cdot\ sin\frac{\pi}{2}}{\ sinx\cdot\ cos\frac{\pi}{2} -\ sin\frac{\pi}{2}\cdot\ cosx} = \frac{\ sinx}{-\cos x}

by simplifying \displaystyle{ \sin \frac{ \pi }{2}=1 and \displaystyle{ \cos \frac{ \pi }{2}=0.

Now, \displaystyle{ \frac{\ sinx}{-\cos x} is clearly -tanx.
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Option B

The number of light years in 6.79 \times 10^{16} miles is 11508 light years

<em><u>Solution:</u></em>

Given that,

One light-year equals 5.9 x 10^12 miles

Therefore,

1 \text{ light year } = 5.9 \times 10^{12} \text{ miles }

To find: Number of light years in 6.79 \times 10^{16} miles

Let "x" be the number of light years in 6.79 \times 10^{16} miles

Then number of light years in 6.79 \times 10^{16} miles can be found by dividing 6.79 \times 10^{16} miles by miles in 1 light year

\text{Number of light years in } 6.79 \times 10^{16} miles = \frac{6.79 \times 10^{16}}{5.9 \times 10^{12}}\\\\\text{Use the law of exponent }\\\\\frac{a^m}{a^n} = a^{m-n}\\\\\text{Number of light years in } 6.79 \times 10^{16} miles = \frac{6.79}{5.9} \times 10^{16-12}\\\\\text{Number of light years in } 6.79 \times 10^{16} miles = 1.1508 \times 10^4\\\\\text{Number of light years in } 6.79 \times 10^{16} miles = 11508

Thus number of light years in 6.79 \times 10^{16} miles is 11508 light years

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