Question

Complete the proof of the identity by choosing the rule that justifies each step of

Answer 1

To verify cos(x-y)/cosxcosy=1+tanxtany we proceed as follows:
cos (x+y)=cos x cos y-sin x sin y
considering left side we get
[cos(x+y)]/[cos x cos y]
=[cos x cos y-sin xsin y]/[cos x cos y]
=1-[sin x sin y]/[cos x cosy]
=1-[sin x/ cos x × sin y/ cos y]
=1-tan x tan y
 Hence the proof

Answer 2

cos ( x - y ) / cos x  cos y = 1 + tan x tan y is proven as explained below.

Further explanation

Firstly , let us learn about trigonometry in mathematics.

Suppose the ΔABC is a right triangle and ∠A is 90°.

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

There are several trigonometric identities that need to be recalled, i.e.

$cosec ~ A = \frac{1}{sin ~ A}$

$sec ~ A = \frac{1}{cos ~ A}$

$cot ~ A = \frac{1}{tan ~ A}$

$tan ~ A = \frac{sin ~ A}{cos ~ A}$

Let us now tackle the problem!

In this problem , we will use identity as follows:

$\large {\boxed {\cos (x - y) = \cos x ~ \cos y + \sin x ~ \sin y } }$

Given:

$(\cos ( x - y )) / (\cos x \cos y) = \frac{(\cos x ~ \cos y + \sin x ~ \sin y)}{(\cos x \cos y)}$

$(\cos ( x - y )) / (\cos x \cos y) = \frac{(\cos x ~ \cos y)}{(\cos x \cos y)} + \frac{(\sin x ~ \sin y)}{(\cos x \cos y)}$

$(\cos ( x - y )) / (\cos x \cos y) = 1 + \frac{(\sin x ~ \sin y)}{(\cos x \cos y)}$

$(\cos ( x - y )) / (\cos x \cos y) = 1 + \frac{(\sin x)}{(\cos x)} \frac{(\sin y)}{(\cos y)}$

$\large {\boxed {(\cos ( x - y )) / (\cos x \cos y) = 1 + \tan x \tan y} }$

Learn more

• Calculate Angle in Triangle : brainly.com/question/12438587
• Periodic Functions and Trigonometry : brainly.com/question/9718382
• Trigonometry Formula : brainly.com/question/12668178

Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse , Triangle , Fraction , Lowest , Function , Angle