R. D. Sharma Solutions for Chapter: Trigonometric Identities, Exercise 3: VERY SHORT ANSWER TYPE QUESTIONS (VSAQs)

If $5\mathrm{x}=\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{\theta }$ and $\frac{5}{\mathrm{x}}=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{\theta },$ find the value of $5\left({\mathrm{x}}^2-\frac{1}{{\mathrm{x}}^2}\right)$.

If $\mathrm{cosec\theta }=2x$ and $\mathrm{c}\mathrm{o}\mathrm{t\theta }=\frac{2}{x}$, find the value of $2\left(x^2-\frac{1}{x^2}\right)$.

The value of $\sin \theta$ is $x + \frac{1}{x}$, where $x$ is a positive real number.

$\cos \theta =\frac{a^2+b^2}{2ab}$, where $a$and $b$are two distinct numbers such that $ab> 0$.

The value of ${\cos }^2 23°-{\sin }^2 67°$ is positive.

The value of the expression $\sin 80°-\cos 80°$ is negative.

The value of $\sin \theta + \cos \theta$ is always greater than $1$.

What is the value of ${\cos }^{2}67°-{\sin }^{2}23°?$