Question

Prove that ${\sin ^4}\theta - {\cos ^4}\theta + 1 = 2{\sin ^2}\theta $

Answer 1

Hint: To solve this question firstly put the value of ${\sin ^2}\theta $ in terms of ${\cos ^2}\theta $ and after that solve the equation and try to make it equal to R.H.S. The value of ${\sin ^2}\theta $ can be obtained from the following trigonometric identity i.e.
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1$

Complete step-by-step answer:
We have to prove
$ \Rightarrow {\sin ^4}\theta - {\cos ^4}\theta + 1 = 2{\sin ^2}\theta $
To prove this equation, we will solve L.H.S and try to make it equal by using the appropriate trigonometric identity.
Solve the L.H.S of the equation i.e.
$ \Rightarrow {\sin ^4}\theta - {\cos ^4}\theta + 1 = {({\sin ^2}\theta )^2} - {\cos ^4}\theta + 1$ …..(1)
Here we will use a trigonometric identity:
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1$
$ \Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta $ …….(2)
Put this value in equation 1 we get,
$ \Rightarrow {\sin ^4}\theta - {\cos ^4}\theta + 1 = {(1 - {\cos ^2}\theta )^2} - {\cos ^4}\theta + 1$
Apply the formula i.e. ${(a - b)^2} = {a^2} + {b^2} - 2ab$ on ${(1 - {\cos ^2}\theta )^2}$we get,
$
   \Rightarrow {\sin ^4}\theta - {\cos ^4}\theta + 1 = 1 + {\cos ^4}\theta - 2{\cos ^2}\theta - {\cos ^4}\theta + 1 \\
   \Rightarrow L.H.S = 1 - 2{\cos ^2}\theta + 1 \\
   \Rightarrow L.H.S = 2 - 2{\cos ^2}\theta \\
$
Taking 2 common from both the terms we get,
$ \Rightarrow L.H.S = 2(1 - {\cos ^2}\theta )$
From 2 we get,
$ \Rightarrow 2(1 - {\cos ^2}\theta ) = 2{\sin ^2}\theta $=R. H. S
Here you can see that L.H.S = R.H.S
Hence, proved.

Note: Some students take $\sin \theta $ as product of $\sin $ and $\theta $. But it is wrong, the $\sin \theta $ means sine of angle $\theta $. These trigonometric functions depend only on the value of the angle $\theta $ and not on the position chosen on the terminal side of the angle $\theta $.
The trigonometric ratios are the same for the same angles. If the terminal sides coincide with x-axis then $\cos ec\theta \& \cot \theta $are not defined and if it coincides with y axis, then $sec\theta \& \tan \theta $ are not defined. The trigonometric ratios can be positive and negative depending upon x and y.
An equation which involves trigonometric functions and is true for all those angles for which functions are defined is called a trigonometric identity. Some basic trigonometric identities are as follows: -
${\sin ^2}\theta + {\cos ^2}\theta = 1$
$1 + {\tan ^2}\theta = {\sec ^2}\theta $
$1 + {\cot ^2}\theta = \cos e{c^2}\theta $