Finding tan theta Value Using P = tan theta + sec theta
We are given the expression $P = \tan\theta + \sec\theta$. Our goal is to find the value of $\tan\theta$ in terms of $P$.
Using Trigonometric Identities
We know the fundamental trigonometric identity: $ \sec^2\theta - \tan^2\theta = 1 $ This can be factored as the difference of squares:
$ (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 $
Substitute the given expression $P = \tan\theta + \sec\theta$ into the factored identity:
$ (\sec\theta - \tan\theta) P = 1 $
Now, we can express $(\sec\theta - \tan\theta)$ in terms of $P$:
$ \sec\theta - \tan\theta = \frac{1}{P} $
Solving for tan theta
We now have a system of two linear equations involving $\sec\theta$ and $\tan\theta$:
- $ \sec\theta + \tan\theta = P $
- $ \sec\theta - \tan\theta = \frac{1}{P} $
To find $\tan\theta$, subtract the second equation from the first:
$ (\sec\theta + \tan\theta) - (\sec\theta - \tan\theta) = P - \frac{1}{P} $
Simplify the left side:
$ \sec\theta + \tan\theta - \sec\theta + \tan\theta = P - \frac{1}{P} $
$ 2\tan\theta = P - \frac{1}{P} $
Combine the terms on the right side using a common denominator:
$ 2\tan\theta = \frac{P \cdot P}{P} - \frac{1}{P} $
$ 2\tan\theta = \frac{P^2 - 1}{P} $
Finally, divide by 2 to isolate $\tan\theta$:
$ \tan\theta = \frac{P^2 - 1}{2P} $
This matches the value provided in Option D.
Similar Questions
- If $\cos(x + y) = \frac{1}{2}$ and $\sin(x - y) = 0$, where x and y are positive acute angles and $x \ge y$, then x and y are:
- If $\tan A = \frac{3}{4}$, then the value of $\frac{\cos A - \sin A}{\cos A + \sin A}$ is:
- What is the value of $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$?
- What is the value of the following expression?
$(\tan 2^\circ \tan 88^\circ) (\tan 3^\circ \tan 87^\circ) ... (\tan 43^\circ \tan 47^\circ) \tan 45^\circ$ - If the ratio of the sine of an acute angle to its cosine is 5 : 12, then what will the value of sine of that angle be?
- If $\tan\theta + \cot\theta = 6$, then find the value of $\tan^2\theta + \cot^2\theta$
- What is the value of the following expression?
$\frac{\cos 3x + \cos x}{\sin 3x - \sin x}$ - If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$ with $0^\circ < (A + B) \leq 90^\circ$, $A > B$, then find the measure of A and B.
- If $\sqrt{2} \sin(5x - 5)^\circ = \tan 45^\circ$, then the value of x (in degrees) is:
- If $\cos 2\theta = \frac{1}{2}$, then the value of $\sin(75^\circ - \theta)$ will be:
Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
