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Title: Trigonometry Identities (quotient , co-function)

Grade: 10-a Lesson: S3-L4

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Use cofunction identities to simplify the expression \$ csc^2 x − tan^2 (pi/2 − x) \$.
2	Step	The given expression is	\$ csc^2 x − tan^2 (pi/2 − x) \$
3	Step	The cofunction identity for tangent is:	\$ tan^2 (pi/2 − x) = cot^2 x \$
4	Step	Rewrite the given expression using the cofunction identity:	\$ csc^2(x) − cot^2(x) \$
5	Formula:	There is a Pythagorean identity relating cosecant and cotangent:	\$ csc^2(x) = 1 + cot^2(x) \$
6	Step	Substitute the identity \$ csc^2(x) = 1+ cot^2(x) \$ into the simplified expression:	\$ (1 + cot^2(x)) − cot^2(x) \$ ⇒ \$ 1 + cot^2(x) − cot^2(x) \$ ⇒ 1
7	Step	Thus, the simplified expression for \$ csc^2 x − tan^2 (pi/2 − x) \$ is 1.
8	Choice.A	The simplified expression is 1, which matches option A	1
9	Choice.B	The expression is incorrect as it stands because it simplifies to 1, not 60	60
10	Choice.C	Option C is wrong as the simplified form doesn’t yield 0 but rather 1	0
11	Choice.D	This choice is incorrect because the simplified expression is not None of the above; it simplifies to 1	None of the above
12	Answer	Option	A
13	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Use cofunction identities to simplify the expression \$ csc^2 x − tan^2 (pi/2 − x) \$.

Step

The given expression is

\$ csc^2 x − tan^2 (pi/2 − x) \$

Step

The cofunction identity for tangent is:

\$ tan^2 (pi/2 − x) = cot^2 x \$

Step

Rewrite the given expression using the cofunction identity:

\$ csc^2(x) − cot^2(x) \$

Formula:

There is a Pythagorean identity relating cosecant and cotangent:

\$ csc^2(x) = 1 + cot^2(x) \$

Step

Substitute the identity \$ csc^2(x) = 1+ cot^2(x) \$ into the simplified expression:

\$ (1 + cot^2(x)) − cot^2(x) \$

⇒ \$ 1 + cot^2(x) − cot^2(x) \$

⇒ 1

Step

Thus, the simplified expression for \$ csc^2 x − tan^2 (pi/2 − x) \$ is 1.

Choice.A

The simplified expression is 1, which matches option A

Choice.B

The expression is incorrect as it stands because it simplifies to 1, not 60

Choice.C

Option C is wrong as the simplified form doesn’t yield 0 but rather 1

Choice.D

This choice is incorrect because the simplified expression is not None of the above; it simplifies to 1

None of the above

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?