Question

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Answer 1

Answer:

2secx

Step-by-step explanation:

$\tan(x) + \sec(x) = p$

$= ( \tan(x) + \sec(x) ) \frac{( \tan(x) - \sec(x) )}{ \tan(x) - \sec(x) ) }$

$= {( \tan(x) )}^{2} - { (\sec(x)) }{2}$

divided by tan x - sec x

nothing but 1÷p is sec x - tan x

so p+ 1÷p is tanx + sec x +sec x -tanx

=2sec x

Answer 2

Note :

Here let's take the symbol theta as @.

i.e. @ = θ ( theta )

Question :

Let p = tan @ + sec @ , then find the value of p + 1/p ?

Answer

Given : -

p = tan @ + sec @

Required to find : -

• Value of p + 1/p ?

Identity used : -

sin² @ + cos² @ = 1

Solution : -

⟶ p = tan @ + sec @

we need to find the value of p + 1/p

So,

Let consider the value of p !

⟶ tan @ + sec @

we know that ;

• tan @ = sin @/cos @

• sec @ = 1/cos @

This implies ;

⟶ sin @/cos @ + 1/cos @

⟶ sin @ + 1/cos @

Hence,

Let's consider the value of p as ,

• p = sin @ + 1/cos @

Similarly,

Now,

Let's try to find the value of 1/p

1/p =

⟶ 1/ sin @ + 1/cos @

⟶ 1/1 ÷ sin @ + 1/cos @

⟶ 1/1 x cos @/sin @ + 1

⟶ cos @/sin @ + 1

Now,

Let's multiply the numerator and denominator with sin @ - 1

⟶ cos @/sin @ + 1 x sin @ - 1/sin @ - 1

⟶ cos @ ( sin @ - 1 )/ ( sin @ )² - ( 1 )²

[ from the identity : ( a + b ) ( a - b ) = a² - b² ]

⟶ cos @ sin @ - cos @/ sin² @ - 1

From the identity ;

sin² @ + cos² @ = 1

sin² @ = 1 - cos² @

⟶ cos @ sin @ - cos @/ 1 - cos² @ - 1

+ 1 , - 1 get's cancelled in the denominator

⟶ cos @ sin @ - cos @/- cos² @

Now,

Let's split this into 2 parts ;

⟶ cos @ sin @/ - cos² @ - [ cos @/- cos² @ ]

⟶ sin @/ - cos @ - [ 1/ - cos @ ]

since,

• sin @/ cos @ = tan @

• 1/ cos @ = sec @

⟶ - tan @ - [ - sec @ ]

⟶ - tan @ + sec @

Hence,

• Value of p + 1/p = - tan @ + sec @

Now,

According to question ;

p + 1/p

⟶ tan @ + sec @ + ( - tan @ + sec @ )

⟶ tan @ + sec @ - tan @ + sec @

tan @ , - tan @ get's eliminated

⟶ sec @ + sec @

⟶ 2 sec @

Additional Information : -

The six trigonometric ratio's are ;

• sin @ = opposite side/Hypotenuse
• cos @ = Adjacent side/Hypotenuse

• tan @ = Opposite side/Adjacent side

• cot @ = Adjacent side/ Opposite side

• sec @ = Hypotenuse/Adjacent side

• cosec @ = Hypotenuse/Opposite side

Moreover,

sin @ = 1/cosec @

cos @ = 1/sec @

tan @ = 1/cot @

These are the reciprocal ratio's .

In trigonometry ,

The interesting thing is that there is only unique solution for a question but we don't have unique method to a solve a question .

Simply ;

We have different methods to solve a single question .

Some formulae are ;

sin ( A + B ) = sin A cos B + sin B cos A

sin ( A - B ) = sin A cos B - sin B cos A

cos ( A + B ) = cos A cos B - sin A sin B

cos ( A - B ) = cos A cos B + sin A sin B

Complementary angles

sin @ = cos ( 90° - @ )

cos @ = sin ( 90° - @ )

tan @ = cot ( 90° - @ )

cot @ = tan ( 90° - @ )

sec @ = cosec ( 90° - @ )

cosec @ = sec @ ( 90° - @ )