Mathematics
19.07.2020 11:44
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Verify that seca + tana = (cos)/(1-sina)

Verify that seca + tana = (cos)/(1-sina)
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Victoriag2626
Victoriag2626
4,6(81 marks)
Refresh if you see display errors such as \begin{aligned}, etc.======\sec a + \tan a = \dfrac{\cos a}{1-\sin a}focusing  on the left-hand side    \text{lhs} = \sec a + \tan aone strategy is to rewrite everything in terms of sine and cosine.since \sec a = \frac{1}{\cos a} and  \tan a = \frac{\sin a}{\cos a}, we have    \begin{aligned} \text{lhs} & = \sec a + \tan a \\ & = \frac{1}{\cos a} + \frac{\sin a}{\cos a} \\ & = \frac{1 + \sin a}{\cos a} \end{aligned}they have the same denominator, which is why i am able to combine them.that looks as simple as simple gets. can leave it alone. (read the remark at the end, because there are things you can do here, too.)======focusing  on the right-hand side    \text{rhs} = \dfrac{\cos a}{1-\sin a}in this situation, there is one strategy we can use: multiplication by the conjugate.for the right hand side, the conjugate of  1-\sin a is  1+\sin a. we can multiply the numerator and denominator of the right hand side by this conjugate.why do this? because it will produce a difference of squares in the denominator, which may permit us to use a pythagorean identity.(another rationale is that, if you look on the left side, we have 1+sina; this can lead to you conclude that we need to multiply by 1+sina on the right side to get it there as well.)    \begin{aligned} \text{rhs} & = \dfrac{\cos a}{1-\sin a} \cdot \frac{1+\sin a}{1+\sin a} \\ & = \dfrac{\cos a(1+\sin a)}{(1-\sin a)(1+\sin a)} \end{aligned}the denominator is a difference of squares expansion, in the form of  (a-b)(a+b) = a^2 - b^2. doing this to the denominator gets us    \begin{aligned} \text{rhs} & = \dfrac{\cos a(1+\sin a)}{(1-\sin a)(1+\sin a)} \\ & = \dfrac{\cos a(1+\sin a)}{1-\sin^2 a} \end{aligned}the denominator is a variation of the pyth. identity.with  \sin^2 a + \cos^2 a = 1, subtracting both sides by  \sin^2 a gets us  \cos^2 a= 1 -\sin^2 a. hence    \begin{aligned} \text{rhs} & = \dfrac{\cos a(1+\sin a)}{1-\sin^2 a} \\ & = \dfrac{\cos a(1+\sin a)}{\cos^2 a} \end{aligned}the cosines (partially) cancel out and we are left with    \begin{aligned} \text{rhs} & = \dfrac{1+\sin a}{\cos a} \\ & = \text{lhs} \end{aligned}======you could have also did the conjugate step for the left-hand side when you finished simplifying it all. there are usually many ways to prove an identity.i hope i'm not doing your entire homework packet or something like that (though i do understand that this one involves something different than the other questions you've posted)
shaviaa
shaviaa
4,4(71 marks)

1.923=p If you are rounding you can round to 1.9

Step-by-step explanation:

We must simply get P alone.

1400 = 2p*52*7

First we can multiply the integers

1400 = 2p*364

To get P alone we can divide by 364 on both sides

1400/364 = 2p

3.846 = 2p

Divide by 2 on both sides

3.846/2 = p

1.923 = p

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