Question

Show that
TanO*SinO/1-CosO=1+1/CosO

Answer 1

you can replace O with x, as it can be thought as Zero (0)..

Answer 2

\[\frac{ \tan \Theta \sin \Theta }{ 1-\cos \Theta? }= 1+\frac{ 1 }{ \cos \Theta }\]

Answer 3

\[\frac{\sin(\theta) \cdot \tan(\theta)}{1 - \cos(\theta)}\]

Answer 4

Okay..

Answer 5

Start by :

\[\tan(\theta) =\frac{\sin(\theta)}{\cos(\theta)}\]

Answer 6

What do you get when you will use this??

Answer 7

\[\frac{ \sin ^{2\Theta} }{ 1-\cos ^{2} \Theta }\]

Answer 8

?

Answer 9

LHS:

\[\implies \frac{\sin^2(\theta)}{\cos(\theta) [1 - \cos(\theta)]}\]

Answer 10

No, the cos(theta) will be multiplied by 1 also, you forgot to do that..

Don't multiply cos(theta) inside the brackets..

Answer 11

\[1 + \cos(\theta) = 2 \cos^2(\frac{\theta}{2})\\1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})\]

Answer 12

For the 1-cos(theta) in denominator, use the above identity..

Answer 13

\[\frac{\sin^2(\theta)}{\cos(\theta)[2 \sin^2(\frac{\theta}{2})]}\]

Answer 14

Now you also know that:

\[\sin(\theta) = 2 \sin(\frac{\theta}{2})\cos(\frac{\theta}{2})\]

Answer 15

u can also use\[\sin^2 \theta(1+\cos \theta)/\cos \theta(1-\cos \theta)(1+\cos \theta)\]

Answer 16

\[\frac{4 \sin^2(\frac{\theta}{2}) \cos^2(\frac{\theta}{2})}{2 \sin^2(\frac{\theta}{2}) \cos(\theta)} \\ \\\]

Answer 17

\[\implies \frac{2\cos^2(\frac{\theta}{2})}{\cos(\theta)} \implies \frac{1 + \cos(\theta)}{\cos(\theta)} \implies \color{blue}{ 1 + \frac{1}{\cos(\theta)}}\]

Answer 18

Oh, I did not see that, you can also try rationalising the denominator..

Answer 19

@mathmath333 method is simpler one..

Answer 20

\[LHS \implies \frac{\sin^2(x)}{\cos(x)[1 - \cos(x)]}\]

I am using x for \(\theta\) here,,

Answer 21

Here multiply and divide by (1 + cos(x))..

\[\frac{\sin^2(x) [1 + \cos(x)]}{\cos(x) (1 - \cos(x))[1 + \cos(x)]} \implies \frac{\sin^2(x) [1 + \cos(x)]}{\cos(x) \cdot \sin^2(x)} \\ \frac{1 + \cos(x)}{\cos(x)} \implies 1 + \frac{1}{\cos(x)}\]

Answer 22

Note that:
In denominator I have done:

\[(1 - \cos(x))(1 + \cos(x)) = 1 - \cos^2(x) = \sin^2(x)\]

Answer 23

Based on what did we multiply and divide by (1+cos(x))?

Answer 24

Logical thinking..

As you have (1- cos(x)) on down, so your mind must tell you that if I multiply it with (1+ cos(x)), you will get sin^2(x) in denominator, and upper sin(x) can get cancelled with it,..

Answer 25

In proving Trigonometric Identities, just start from the identity you know, there is nothing hard and fast rule, from all the methods you will get the same answer, some method can take longer steps like MINE, and some can take smaller like @mathmath333

Answer 26

Just start by what you know, you will definitely reach to your answer, but while proving, do look at your RHS, as you want to make your LHS equals RHS..

So, you can't do anything that will take away from your RHS.. You will get it, solve like 50 of these problems, you will gain confidence.. :)

Answer 27

I gotta go now..

@mathmath333 will take on now onwards.. :)

Answer 28

Thank you :)

Answer 29

any problem @Attyeh

Answer 30

for expertise in trignometry u should remember and try to prove all trignometric identities ,formulas

Answer 31

I do face a problem regarding trigonometry, do you have a list of the most needed identities? Im doin P1 in October and im trying to Ace it so any help would be appreciated
@mathmath333