Is 7csc^2(x)-5cot^2(x)=2csc^2(x)+5 ?

The answer to whether 7csc^2(x)-5cot^2(x)=2csc^2(x)+5 is True

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prove 7csc^2(x)-5cot^2(x)=2csc^2(x)+5

prove

7 csc^{2} (x) - 5 cot^{2} (x) = 2 csc^{2} (x) + 5

True

Solution steps

7 csc^{2} (x) - 5 cot^{2} (x) = 2 csc^{2} (x) + 5

Manipulating left side

7 csc^{2} (x) - 5 cot^{2} (x)

Rewrite using trig identities

7 csc^{2} (x) - 5 cot^{2} (x)

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= 7 csc^{2} (x) - 5 (csc^{2} (x) - 1)

Simplify

7 csc^{2} (x) - 5 (csc^{2} (x) - 1) : 2 csc^{2} (x) + 5

7 csc^{2} (x) - 5 (csc^{2} (x) - 1)

Expand

- 5 (csc^{2} (x) - 1) : - 5 csc^{2} (x) + 5

- 5 (csc^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - 5, b = csc^{2} (x), c = 1

= - 5 csc^{2} (x) - (- 5) \cdot 1

Apply minus-plus rules

- (- a) = a

= - 5 csc^{2} (x) + 5 \cdot 1

Multiply the numbers:

5 \cdot 1 = 5

= - 5 csc^{2} (x) + 5

= 7 csc^{2} (x) - 5 csc^{2} (x) + 5

Add similar elements:

7 csc^{2} (x) - 5 csc^{2} (x) = 2 csc^{2} (x)

= 2 csc^{2} (x) + 5

= 2 csc^{2} (x) + 5

= 2 csc^{2} (x) + 5

We showed that the two sides could take the same form

\Rightarrow True

Is 7csc^2(x)-5cot^2(x)=2csc^2(x)+5 ?
The answer to whether 7csc^2(x)-5cot^2(x)=2csc^2(x)+5 is True