Question

How to solve trigonometric questions easily

Answer 1

by the very easy way

Answer 2

PRACTICE MAKES A MAN PERFECT

here are some points which will help you

Quotient Identities:

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)

Reciprocal Identities:

cosec(x) = 1/sin(x)

sec(x) = 1/cos(x)

cot(x) = 1/tan(x)

sin(x) = 1/cosec(x)

cos(x) = 1/sec(x)

tan(x) = 1/cot(x)

Pythagorean Identities:

sin2(x) + cos2(x) = 1

cot2A +1 = cosec2A

1+tan2A = sec2A

How to Solve Them Correctly Every Time

STEP 1: Convert all sec, cosec, cot, and tan to sin and cos. Most of this can be done using the quotient and reciprocal identities.

STEP 2: Check all the angles for sums and differences and use the appropriate identities to remove them.

STEP 3: Check for angle multiples and remove them using the appropriate formulas.

STEP 4: Expand any equations you can, combine like terms, and simplify the equations.

STEP 5: Replace cos powers greater than 2 with sin powers using the Pythagorean identities.

STEP 6: Factor numerators and denominators, then cancel any common factors.

STEP 7: Now, both sides should be exactly equal, or obviously equal, and you have proven your identity.

example using the steps

Show that cos4(x) – sin4(x) = cos(2x)

STEP 1: Everything is already in sin and cos, so this part is done.

cos4(x) – sin4(x) = cos (2x)

STEP 2: Since there are no sums or difference inside the angles, this part is done.

cos4(x) – sin4(x) = cos (2x)

STEP 3: cos(2x) is a double angle. Use the double angle formula: cos (2x) = cos2(x) – sin2(x), to simplify.

cos4(x) – sin4(x) = cos2(x) – sin2(x)

STEP 4: Here is where your algebra knowledge comes in. In this case, we can see that the left side is a “difference of two squares”

[if you forgot: a2-b2 = (a+b)(a-b)]

Left side: cos4x – sin4x – (cos2(x))2 – (cos2(x))2 = (cos2(x)-sin2(x))(cos2(x)+sin2(x))

Now, our problem looks like this:

(cos2(x)-sin2x))(cos2(x)+sin2(x))= cos2(x) – sin2(x)

The sides are almost the same

STEP 5: There are no powers greater than 2, so we can skip this step

STEP 6: Since cos2(x) – sin2(x) appears on both sides of the equation, we can cancel it.

We are left with: cos2(x) + sin2(x) = 1

STEP 7: Since this is one of the pythagorean identities, we know it is true, and the problem is done.

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