# Verify the following identity: `tanx(cot x + tan x) = sec^2x`

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We have to verify that `tanx(cot x + tan x) = sec^2x`

=> `tan x*cot x+ tan^2x`

=> `1 + tan^2x`

=> `1 + (sin^2x)/(cos^2x)`

=> `(cos^2x + sin^2x)/(cos^2x)`

=` `> `1/(cos^2x)`

=> `sec^2x`

This proves that `tan x(cot x + tan x) = sec^2 x`

Posted on

An alternative method is to take `sec^2 x ` and replace it by `1/(cos^2 x).`

Replace `1/(cos^2 x)`  by `1+ tan^2 x`  (the basic formula of trigonometry `1+tan^2 x = 1/(cos^2 x)` ).

Use this substitution you've made in the identity to be proven.

`tan x(cot x + tan x) = 1+tan^2 x`

Opening the brackets, you'll have`tanx*cotx + tan^2x = 1+tan^ 2 x`

The cotangent function is the inverse of tangent, therefore tanx*cot x = 1.

`1+ tan^2x = 1+tan^ 2 x`

ANSWER: The last line proves the identity`tan x(cot x + tan x) = sec^2 x`

Posted on

L:H:S ≡ tanx(cotx + tanx)

= tanx.cotx + tan²x

⇒ tanθ.cotθ = 1

= 1 + tan²x

⇒ 1 + tan²θ = sec²θ

= sec²x

= R:H:S
The text passage contains a mathematical identity to be verified, along with multiple methods of proving the identity. The identity is `tanx(cot x + tan x) = sec^2x`, and the passage provides step-by-step solutions to prove this identity.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Verify the following identity: `tanx(cot x + tan x) = sec^2x`

Posted on

We have to verify that `tanx(cot x + tan x) = sec^2x`

=> `tan x*cot x+ tan^2x`

=> `1 + tan^2x`

=> `1 + (sin^2x)/(cos^2x)`

=> `(cos^2x + sin^2x)/(cos^2x)`

=` `> `1/(cos^2x)`

=> `sec^2x`

This proves that `tan x(cot x + tan x) = sec^2 x`

Posted on

An alternative method is to take `sec^2 x ` and replace it by `1/(cos^2 x).`

Replace `1/(cos^2 x)`  by `1+ tan^2 x`  (the basic formula of trigonometry `1+tan^2 x = 1/(cos^2 x)` ).

Use this substitution you've made in the identity to be proven.

`tan x(cot x + tan x) = 1+tan^2 x`

Opening the brackets, you'll have`tanx*cotx + tan^2x = 1+tan^ 2 x`

The cotangent function is the inverse of tangent, therefore tanx*cot x = 1.

`1+ tan^2x = 1+tan^ 2 x`

ANSWER: The last line proves the identity`tan x(cot x + tan x) = sec^2 x`

Posted on

L:H:S ≡ tanx(cotx + tanx)

= tanx.cotx + tan²x

⇒ tanθ.cotθ = 1

= 1 + tan²x

⇒ 1 + tan²θ = sec²θ

= sec²x

= R:H:S
user
The text passage contains a mathematical identity to be verified, along with multiple methods of proving the identity. The identity is `tanx(cot x + tan x) = sec^2x`, and the passage provides step-by-step solutions to prove this identity.

TYPE: SINGLE_QUESTION_WITH_ANSWER