Question

solve and answer it exactly.

√(1+tan²C)​

Answer 1

We need to recall the following trigonometric formula.

• $1+tan^2x=sec^2x$

This problem is about the trigonometric formula.

Given:

$\sqrt{(1+tan^2C)}$

Using the above trigonometric formula, we get

$\sqrt{(1+tan^2C)}=\sqrt{sec^2C}$

                      $=sec(C)$

Thus, $\sqrt{(1+tan^2C)}=secC$

Answer 2

The value of the given expression √(1 + tan²C) is equal to sec(C).

Given:

The expression is √(1 + tan²C).

To Find:

The value of the given expression.

Solution:

→ The tan function is the ratio of perpendicular and base whereas the sec function is the ratio of hypotenuse and base.

              $\boldsymbol{\therefore tanA=\frac{Perpendicular}{Base} }\:\:\:\boldsymbol{and\:\:\:\:secA=\frac{Hypotenuse}{Base} }$

→ Therefore tanC would be equal to the ratio of Perpendicular and Base.

                                  $\therefore \sqrt{1+tan^{2}C } \\\\= \sqrt{1+(\frac{Perpendicular}{Base} )^{2} }\\\\=\sqrt{(\frac{(Base)^{2} +(Perpendicular)^{2} }{(Base)^{2} } )}$

→ We know by Pythagoras's Theorem:

                    (Hypotenuse)² = (Base)² + (Perpendicular)²

→ The given expression simplifies to:

                                 $=\sqrt{(\frac{(Base)^{2} +(Perpendicular)^{2} }{(Base)^{2} } )}\\\\=\sqrt{(\frac{(Hypotenuse)^{2} }{(Base)^{2} } )}\\\\=\sqrt{(\frac{Hypotenuse}{Base} )^{2} } \\\\=\frac{Hypotenuse}{Base} \\\\=sec(C)$

Therefore the value of the given expression √(1 + tan²C) comes out to be equal to sec(C).

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