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Exercise

$\frac{\sqrt{\sec^2\left(x\right)-49}}{7\sec\left(x\right)}$

Step-by-step Solution

1

Applying the trigonometric identity: $\sec\left(\theta \right)^2 = 1+\tan\left(\theta \right)^2$

$\frac{\sqrt{-48+\tan\left(x\right)^2}}{7\sec\left(x\right)}$
Why is tan(x)^2+1 = sec(x)^2 ?
2

Rewrite $\frac{\sqrt{-48+\tan\left(x\right)^2}}{7\sec\left(x\right)}$ in terms of sine and cosine functions

$\frac{\sqrt{-48+\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}}}{\frac{7}{\cos\left(x\right)}}$
3

Combine $-48+\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}$ in a single fraction

$\frac{\sqrt{\frac{\sin\left(x\right)^2-48\cos\left(x\right)^2}{\cos\left(x\right)^2}}}{\frac{7}{\cos\left(x\right)}}$
4

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{\frac{\sqrt{\sin\left(x\right)^2-48\cos\left(x\right)^2}}{\cos\left(x\right)}}{\frac{7}{\cos\left(x\right)}}$
5

Simplify the fraction $\frac{\frac{\sqrt{\sin\left(x\right)^2-48\cos\left(x\right)^2}}{\cos\left(x\right)}}{\frac{7}{\cos\left(x\right)}}$

$\frac{\sqrt{\sin\left(x\right)^2-48\cos\left(x\right)^2}}{7}$

Final answer to the exercise

$\frac{\sqrt{\sin\left(x\right)^2-48\cos\left(x\right)^2}}{7}$

Try other ways to solve this exercise

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Similar Questions Solved

$\left(\sin\left(x\right)+\cos\left(x\right)\right)^2$

$1+cot^2x$

$tan\left(x\right)csc\left(x\right)$

$\tan\left(x\right)\csc\left(x\right)$

$\frac{2+\tan^2\left(x\right)}{\sec^2\left(x\right)}-1$

$csc^2x-1$

$\frac{2+tan^2x}{sec^2x}-1$

Main Topic: Simplify Trigonometric Expressions

Simplification of trigonometric expressions consists of rewriting an expression with trigonometric functions in a simpler form. To perform this task, we usually use the most common trigonometric identities, and some algebra.

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