Question

verify the identity
cosθ+sinθ tanθ=secθ
please help will fan and medal

Answer 1

do you have any options to choose from?

Answer 2

no

Answer 3

can someone plz explain how to do this

Answer 4

Im not really good at explaining im better at showing you how to but this is the website that i used for help when i was doing the same stuff. http://www.purplemath.com/modules/proving.htm

Answer 5

ok thanks

Answer 6

yep

Answer 7

hint:
we can rewrite the left side, like below:
\[\cos \theta + \sin \theta \tan \theta = \cos \theta + \sin \theta \cdot \frac{{\sin \theta }}{{\cos \theta }}\]

Answer 8

then i simplfy

Answer 9

yes, please rewrite that quantity as a fraction:
\[\cos \theta + \sin \theta \tan \theta = \cos \theta + \sin \theta \cdot \frac{{\sin \theta }}{{\cos \theta }} = \frac{{...?}}{{\cos \theta }}\]

Answer 10

tantheta

Answer 11

hint:
we can write this:
\[\begin{gathered}
\cos \theta + \sin \theta \tan \theta = \cos \theta + \sin \theta \cdot \frac{{\sin \theta }}{{\cos \theta }} = \hfill \\
\hfill \\
= \frac{{\cos \theta \cdot \cos \theta + \sin \theta \cdot \sin \theta }}{{\cos \theta }} \hfill \\
\end{gathered} \]

Answer 12

since the common denominator is \(\cos \theta\)

Answer 13

now, you have to apply a fundamental identity of trigonometry, in order to simplify the numerator. What is such identity?

Answer 14

true

Answer 15

i suck at trig

Answer 16

Does verify mean prove the identity? You can verify that it works by plugging in \(\theta = 0\) for instance. But I guess it really boils down to what do they actually mean? I'm not a mind reader.

Answer 17

i have prove it is an identity or not

Answer 18

by definition \(\sin \theta\) and \(cos \theta\) are the y-coordinate and x-coordinate, respectively of the point \(P\) along the trigonometric circumference:

the radius of the trigonometric circumference is 1

Answer 19

Now if I apply the theorem of Pitagora to the triangle of my drawing, I can write this:
\(x^2+y^2=1\)
am I right?

Answer 20

yes

Answer 21

ok! Now, please replace \(x\) with \(\cos \theta\), and \(y\) with \(\sin \theta\), what do you get?

Answer 22

i get 1

Answer 23

hint:
if I replace \(x\) with \(\cos \theta\), I get:
\[{\left( {\cos \theta } \right)^2} + {y^2} = 1\]
am I right?

Answer 24

right

Answer 25

now, do the same, namely, replace \(y\) with \(\sin \theta\) and rewrite the equation

Answer 26

cos^2+sin^2=1

Answer 27

correct! What you got is the fundamental identity of trigonometry

Answer 28

so my equation would be a true idenitity

Answer 29

please wait, now use that fundamental identity, in order to simplify the right side:
\[\frac{{\cos \theta \cdot \cos \theta + \sin \theta \cdot \sin \theta }}{{\cos \theta }} = ...?\]

Answer 30

sectheta

Answer 31

correct!

Answer 32

so would this make my identity true

Answer 33

yes!

Answer 34

thank you for your help i really do appreciate it

Answer 35

:)