Question

tan theta(1+sec2 theta)=tan 2theta proof ​

Answer 1

I hope this usefull of you

Answer 2

Answer:

To prove that the following identity is true, we will manipulate the left side of the identity:

1.) tan² θ + 1 = sec² θ to make it look identical to the right side as follows:

2.) (tan θ)² + 1 = sec² θ

Utilizing one of the basic trigonometric identities: tan θ = sin θ ∕ cos θ, we now substitute it into the left side of the above identity and simplify as follows:

3.) (sin θ ∕ cos θ)² + 1 = sec² θ

4.) (sin² θ ∕ cos² θ) + 1 = sec² θ

Now, combining the two terms on the left side by using the fact that the lowest common denominator = cos² θ and that 1 = cos² θ ∕ cos² θ, we have:

5.) (sin² θ ∕ cos² θ) + (cos² θ ∕ cos² θ) = sec² θ

6.) (sin² θ + cos² θ) ∕ cos² θ = sec² θ

Utililizing the basic identity: sin² θ + cos² θ =1 and substituting into the left side, we have:

7.) 1 ∕ cos² θ = sec² θ

8.) (1 ∕ cos θ)² = sec² θ

Since sec θ = 1 ∕ cos θ, we can substitute and simplify on the left side as follows:

9.) (sec θ)² = sec² θ

10.) sec² θ = sec² θ

Therefore, we have now arrived at the desired result on the left side, thus showing why the left side of our original identity equals the right side; therefore, we have also verified our original identity.