Question

How do you find the value of $\tan 20(\theta )$ using the double angle identity?

Answer 1

Hint: First we will evaluate the right-hand of equation and then further the left-hand side of the equation. We will use the following formula $\tan 2\theta = \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}\,\,$ to evaluate and then we will further simplify this expression form and hence evaluate the value of the term.

Complete step by step answer:
We will start off by using the formula
$\tan 2\theta = \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}\,\,$.
Here, we will start by evaluating the right-hand side of the equation.
Hence, the equation will become,
$
   = \tan 20(\theta ) \\
   = \tan (2 \times 10\theta ) \\
   = \dfrac{{2\tan (10\theta )}}{{1 - {{\tan }^2}(10\theta )}} \\
   = \dfrac{{2\tan (2 \times 5\theta )}}{{1 - {{\left( {{{\tan }^2}(2 \times 5\theta )} \right)}^2}}} \\
 $
Hence, the value of the expression $\tan 20(\theta )$ is $\dfrac{{2\tan (2 \times 5\theta )}}{{1 - {{\left( {{{\tan }^2}(2 \times 5\theta )} \right)}^2}}}$.

Additional information: Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant and cosecant functions. They are also termed as arcus functions, anti trigonometric functions or cyclometric functions. If an inverse function exists for a given function $f$, then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by $f$. There is a symmetry between a function and its inverse.

Note: While choosing the side to solve, always choose the side where you can directly apply the trigonometric identities. Also, remember the trigonometric identities ${\sin ^2}x + {\cos ^2}x = 1$ and $\cos 2x = 2{\cos ^2}x - 1$. While opening the brackets make sure you are opening the brackets properly with their respective signs. Also remember that $\tan x = \,\dfrac{{\sin x}}{{\cos x}}$.
While applying the double angle identities, first choose the identity according to the terms you have then choose the terms from the expression involving which you are using the double angle identities. While modifying any identity make sure that when you back trace the identity, you get the same original identity.