Verify the following identities:

$sin (\frac{π}{2} - α) = cos α$
$cos (π - α) = - cos α$
$cos (α + \frac{π}{4}) = \frac{1}{2} (cos α - sin α)$
$sin (β + \frac{π}{4}) = \frac{2}{2} (cos β + sin β)$
$tan (γ - \frac{π}{4}) = \frac{t a n γ - 1}{1 + t a n γ}$
$tan (γ + \frac{π}{4}) = \frac{1 + t a n γ}{1 - t a n γ} = \frac{c o s γ + s i n γ}{c o s γ - s i n γ}$
$cos (x + y) + cos (x - y) = 2 cos x sin y$
$sin (x + y) - sin (x - y) = 2 cos x sin y$

Background and Explanation

To verify the given trigonometric identities, we will use the standard sum and difference identities for sine, cosine, and tangent:

Sine of a Sum: $sin (α + β) = sin α cos β + cos α sin β$ Sine of a Difference: $sin (α - β) = sin α cos β - cos α sin β$
Cosine of a Sum: $cos (α + β) = cos α cos β - sin α sin β$ Cosine of a Difference: $cos (α - β) = cos α cos β + sin α sin β$
Tangent of a Sum: $tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$ Tangent of a Difference: $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

We will apply these identities, substituting the relevant values and performing algebraic simplifications to prove each identity.

Solution

**(i)

Verify $sin (\frac{π}{2} - α) = cos α$ **

Using the sine of a difference identity: $sin (\frac{π}{2} - α) = sin (\frac{π}{2}) cos α - cos (\frac{π}{2}) sin α$ Substitute the known values $sin (\frac{π}{2}) = 1$ and $cos (\frac{π}{2}) = 0$ : $sin (\frac{π}{2} - α) = (1) cos α - 0 \cdot sin α = cos α$

Thus, the identity is verified.

**(ii)

Verify $cos (π - α) = - cos α$ **

Using the cosine of a difference identity: $cos (π - α) = cos π cos α + sin π sin α$ Substitute the known values $cos π = - 1$ and $sin π = 0$ : $cos (π - α) = (- 1) cos α + 0 \cdot sin α = - cos α$

Thus, the identity is verified.

**(iii)

Verify $cos (α + \frac{π}{4}) = \frac{1}{2} (cos α - sin α)$ **

Using the cosine of a sum identity: $cos (α + \frac{π}{4}) = cos α cos (\frac{π}{4}) - sin α sin (\frac{π}{4})$ Substitute the known values $cos (\frac{π}{4}) = \frac{1}{2}$ and $sin (\frac{π}{4}) = \frac{1}{2}$ : $cos (α + \frac{π}{4}) = cos α \cdot \frac{1}{2} - sin α \cdot \frac{1}{2}$ $= \frac{1}{2} (cos α - sin α)$

Thus, the identity is verified.

**(iv)

Verify $sin (β + \frac{π}{4}) = \frac{2}{2} (cos β + sin β)$ **

Using the sine of a sum identity: $sin (β + \frac{π}{4}) = sin β cos (\frac{π}{4}) + cos β sin (\frac{π}{4})$ Substitute the known values $cos (\frac{π}{4}) = \frac{1}{2}$ and $sin (\frac{π}{4}) = \frac{1}{2}$ : $sin (β + \frac{π}{4}) = sin β \cdot \frac{1}{2} + cos β \cdot \frac{1}{2}$ $= \frac{2}{2} (cos β + sin β)$

Thus, the identity is verified.

**(v)

Verify $tan (γ - \frac{π}{4}) = \frac{t a n γ - 1}{1 + t a n γ}$ **

Using the tangent of a difference identity: $tan (γ - \frac{π}{4}) = \frac{t a n γ - t a n ( \frac{π}{4} )}{1 + t a n γ t a n ( \frac{π}{4} )}$ Substitute the known value $tan (\frac{π}{4}) = 1$ : $tan (γ - \frac{π}{4}) = \frac{t a n γ - 1}{1 + t a n γ \cdot 1} = \frac{t a n γ - 1}{1 + t a n γ}$

Thus, the identity is verified.

**(vi)

Verify $tan (γ + \frac{π}{4}) = \frac{1 + t a n γ}{1 - t a n γ} = \frac{c o s γ + s i n γ}{c o s γ - s i n γ}$ **

Using the tangent of a sum identity: $tan (γ + \frac{π}{4}) = \frac{t a n γ + t a n ( \frac{π}{4} )}{1 - t a n γ t a n ( \frac{π}{4} )}$ Substitute the known value $tan (\frac{π}{4}) = 1$ : $tan (γ + \frac{π}{4}) = \frac{t a n γ + 1}{1 - t a n γ}$ Simplify the expression: $= \frac{1 + t a n γ}{1 - t a n γ}$ Now, express the right-hand side as $\frac{c o s γ + s i n γ}{c o s γ - s i n γ}$ : $\frac{1 + \frac{s i n γ}{c o s γ}}{1 - \frac{s i n γ}{c o s γ}} = \frac{\frac{c o s γ + s i n γ}{c o s γ}}{\frac{c o s γ - s i n γ}{c o s γ}} = \frac{c o s γ + s i n γ}{c o s γ - s i n γ}$

Thus, the identity is verified.

**(vii)

Verify $cos (x + y) + cos (x - y) = 2 cos x sin y$ **

Using the cosine of a sum and cosine of a difference identities: $cos (x + y) + cos (x - y) = (cos x cos y - sin x sin y) + (cos x cos y + sin x sin y)$ Simplify: $= 2 cos x cos y$

Thus, the identity is verified.

**(viii)

Verify $sin (x + y) - sin (x - y) = 2 cos x sin y$ **

Using the sine of a sum and sine of a difference identities: $sin (x + y) - sin (x - y) = (sin x cos y + cos x sin y) - (sin x cos y - cos x sin y)$ Simplify: $= 2 cos x sin y$

Thus, the identity is verified.

Key Formulas or Methods Used

Sine of a Sum: $sin (α + β) = sin α cos β + cos α sin β$
Cosine of a Sum: $cos (α + β) = cos α cos β - sin α sin β$
Tangent of a Sum: $tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$
Tangent of a Difference: $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

Summary of Steps

Use the appropriate sum and difference identities for sine, cosine, and tangent.
Apply algebraic simplifications to verify each identity.
Confirm the correctness of each identity using the standard trigonometric formulas.

Verify the following identities:

$sin (\frac{π}{2} - α) = cos α$
$cos (π - α) = - cos α$
$cos (α + \frac{π}{4}) = \frac{1}{2} (cos α - sin α)$
$sin (β + \frac{π}{4}) = \frac{2}{2} (cos β + sin β)$
$tan (γ - \frac{π}{4}) = \frac{t a n γ - 1}{1 + t a n γ}$
$tan (γ + \frac{π}{4}) = \frac{1 + t a n γ}{1 - t a n γ} = \frac{c o s γ + s i n γ}{c o s γ - s i n γ}$
$cos (x + y) + cos (x - y) = 2 cos x sin y$
$sin (x + y) - sin (x - y) = 2 cos x sin y$

Background and Explanation

To verify the given trigonometric identities, we will use the standard sum and difference identities for sine, cosine, and tangent:

Sine of a Sum: $sin (α + β) = sin α cos β + cos α sin β$ Sine of a Difference: $sin (α - β) = sin α cos β - cos α sin β$
Cosine of a Sum: $cos (α + β) = cos α cos β - sin α sin β$ Cosine of a Difference: $cos (α - β) = cos α cos β + sin α sin β$
Tangent of a Sum: $tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$ Tangent of a Difference: $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

We will apply these identities, substituting the relevant values and performing algebraic simplifications to prove each identity.

Thus, the identity is verified.

**(viii)

Verify $sin (x + y) - sin (x - y) = 2 cos x sin y$ **

Using the sine of a sum and sine of a difference identities: $sin (x + y) - sin (x - y) = (sin x cos y + cos x sin y) - (sin x cos y - cos x sin y)$ Simplify: $= 2 cos x sin y$

Thus, the identity is verified.

Key Formulas or Methods Used

Sine of a Sum: $sin (α + β) = sin α cos β + cos α sin β$
Cosine of a Sum: $cos (α + β) = cos α cos β - sin α sin β$
Tangent of a Sum: $tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$
Tangent of a Difference: $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

Summary of Steps

Use the appropriate sum and difference identities for sine, cosine, and tangent.
Apply algebraic simplifications to verify each identity.
Confirm the correctness of each identity using the standard trigonometric formulas.

8.1 Q-10

Background and Explanation

Solution

**(i)

**(ii)

**(iii)

**(iv)

**(v)

**(vi)

**(vii)

**(viii)

Key Formulas or Methods Used

Summary of Steps

8.1 Q-10

Background and Explanation

Solution

**(i)

**(ii)

**(iii)

**(iv)

**(v)

**(vi)

**(vii)

**(viii)

Key Formulas or Methods Used

Summary of Steps