1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lions [1.4K]
3 years ago
5

Complete the identity. 1) sec^4 x + sec^2 x tan^2 x - 2 tan^4 x = ?

Mathematics
1 answer:
Alecsey [184]3 years ago
8 0

Answer:

See Explanation

Step-by-step explanation:

<em>Question like this are better answered if there are list of options; However, I'll simplify as far as the expression can be simplified</em>

Given

sec^4 x + sec^2 x tan^2 x - 2 tan^4 x

Required

Simplify

(sec^2 x)^2 + sec^2 x tan^2 x - 2 (tan^2 x)^2

Represent sec^2x with a

Represent tan^2x with b

The expression becomes

a^2 + ab- 2 b^2

Factorize

a^2 + 2ab -ab- 2 b^2

a(a + 2b) -b(a+ 2 b)

(a -b) (a+ 2 b)

Recall that

a = sec^2x

b = tan^2x

The expression (a -b) (a+ 2 b) becomes

(sec^2x -tan^2x) (sec^2x+ 2 tan^2x)

..............................................................................................................................

In trigonometry

sec^2x =1  +tan^2x

Subtract tan^2x from both sides

sec^2x - tan^2x =1  +tan^2x - tan^2x

sec^2x - tan^2x =1

..............................................................................................................................

Substitute 1 for sec^2x - tan^2x in (sec^2x -tan^2x) (sec^2x+ 2 tan^2x)

(1) (sec^2x+ 2 tan^2x)

Open Bracket

sec^2x+ 2 tan^2x ------------------This is an equivalence

(secx)^2+ 2 (tanx)^2

Solving further;

................................................................................................................................

In trigonometry

secx = \frac{1}{cosx}

tanx = \frac{sinx}{cosx}

Substitute the expressions for secx and tanx

................................................................................................................................

(secx)^2+ 2 (tanx)^2 becomes

(\frac{1}{cosx})^2+ 2 (\frac{sinx}{cosx})^2

Open bracket

\frac{1}{cos^2x}+ 2 (\frac{sin^2x}{cos^2x})

\frac{1}{cos^2x}+ \frac{2sin^2x}{cos^2x}

Add Fraction

\frac{1 + 2sin^2x}{cos^2x} ------------------------ This is another equivalence

................................................................................................................................

In trigonometry

sin^2x + cos^2x= 1

Make sin^2x the subject of formula

sin^2x= 1  - cos^2x

................................................................................................................................

Substitute the expressions for 1  - cos^2x for sin^2x

\frac{1 + 2(1  - cos^2x)}{cos^2x}

Open bracket

\frac{1 + 2  - 2cos^2x}{cos^2x}

\frac{3  - 2cos^2x}{cos^2x} ---------------------- This is another equivalence

You might be interested in
Help needed pleaseee
ohaa [14]

Answer:

A

Step-by-step explanation:

Since we're dealing with perpendicular lines, we take the recipricol of the m value of the original equation.

So the slope we're dealing with is 1/2.

And the equation we have so far is

y = 1/2x + b

Now we solve for b by plugging in the point (4,2).

2 = 1/2(4) + b

2 = 2 + b

b = 0

So the final equation is

y = 1/2x

8 0
2 years ago
Challenge question<br>please help<br>​
Pani-rosa [81]

Answer:

The 3 possible values of x are;

6, 4 and 2

Step-by-step explanation:

If the triangle is isosceles, then two of the sides must be equal

So we equate the sides, 2 at a time to get the different values of x

3x + 4 = 2x + 10

3x-2x = 10-4

x = 6

3x + 4 = x + 12

3x-x = 12-4

2x = 8

x = 8/2 = 4

2x + 10 = x + 12

2x - x = 12-10

x = 2

7 0
3 years ago
If 30g of sugar is needed to make 4 cakes how much grams of sugar is needed to make 7 cakes
arsen [322]
There's two ways to do this.

The first way:
Work out the amount of sugar needed to make one cake, and then multiply that by 7.

30 grams ÷ 4 cakes = 7.5 grams for 1 cake.
7.5 grams × 7 cakes = 52.5 grams for 7 cakes.

The second way:
Work out the ratio of sugar:cake by doing 7 ÷ 4, and then multiplying that value by 30.

7 cakes ÷ 4 cakes = 1.75
1.75 × 30 grams = 52.5 grams for 7 cakes. 

Either way, the answer is 52.5 grams.
6 0
3 years ago
Find the domain for the rational function f of x equals quantity x end quantity divided by quantity x minus 5 end quantity.
Gennadij [26K]

Answer:

All real numbers except for 5.

Step-by-step explanation:

f(x)=\frac{x}{x-5}

The domain of rational functions is determined by the denominator. The denominator cannot equal zero since if they do, the function will be undefined.

Thus, we need to find the zero(s) of the denominator to determine the domain.

x-5=0

x=5

Therefore, the domain of the rational function is all real numbers except for 5.

In set builder notation, this is:

\{x|x\in \mathbb{R}, x\neq 5\}

5 0
2 years ago
True or False?
Daniel [21]

Answer:

True

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • Hadley paddled a canoe 2/3 mile in 1/4 hour. How fast did Hadley paddle in miles per hour ?
    6·1 answer
  • the admission fee for a charity event is 7 dollars for children and 10 dollars for adults the event was attended by 700 people a
    10·1 answer
  • A class has 27 students. Fifteen are boys and the rest are girls . What is the ratio of girls to total students?
    12·1 answer
  • The sum of -8 and a number is at least 9
    9·1 answer
  • Samuel used 1/5 of an ounce of butter to make 1/25 of pound of jelly how many ounces of butter is there per pound of jelly
    12·1 answer
  • Can someone help me with this one?
    7·1 answer
  • What must be added to 3x -7 to make x2 + 4x -1?
    15·2 answers
  • On a​ map, 1 inch equals miles. Two houses are inches apart on the map. What is the actual distance between the​ houses? Use pen
    14·1 answer
  • Write each power of ten in standard notation. 10‐³​
    12·1 answer
  • Find the following measure for this figure.
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!