Is 5sec^2(x)+4=9sec^2(x)-4tan^2(x) ?

The answer to whether 5sec^2(x)+4=9sec^2(x)-4tan^2(x) is True

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prove 5sec^2(x)+4=9sec^2(x)-4tan^2(x)

prove

5 sec^{2} (x) + 4 = 9 sec^{2} (x) - 4 tan^{2} (x)

True

Solution steps

5 sec^{2} (x) + 4 = 9 sec^{2} (x) - 4 tan^{2} (x)

Manipulating right side

9 sec^{2} (x) - 4 tan^{2} (x)

Rewrite using trig identities

9 sec^{2} (x) - 4 tan^{2} (x)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= 9 sec^{2} (x) - 4 (sec^{2} (x) - 1)

Simplify

9 sec^{2} (x) - 4 (sec^{2} (x) - 1) : 5 sec^{2} (x) + 4

9 sec^{2} (x) - 4 (sec^{2} (x) - 1)

Expand

- 4 (sec^{2} (x) - 1) : - 4 sec^{2} (x) + 4

- 4 (sec^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = sec^{2} (x), c = 1

= - 4 sec^{2} (x) - (- 4) \cdot 1

Apply minus-plus rules

- (- a) = a

= - 4 sec^{2} (x) + 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= - 4 sec^{2} (x) + 4

= 9 sec^{2} (x) - 4 sec^{2} (x) + 4

Add similar elements:

9 sec^{2} (x) - 4 sec^{2} (x) = 5 sec^{2} (x)

= 5 sec^{2} (x) + 4

= 5 sec^{2} (x) + 4

= 5 sec^{2} (x) + 4

We showed that the two sides could take the same form

\Rightarrow True

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sin (x) sin (2 x) + cos (x) cos (2 x) = cos (x)

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Is 5sec^2(x)+4=9sec^2(x)-4tan^2(x) ?
The answer to whether 5sec^2(x)+4=9sec^2(x)-4tan^2(x) is True